Exercises in Math 110: Algebra and Trigonometry for Calculus

Exercises in Math 110:Algebra and Trigonometry

for CalculusSpring 2019 Edition

Department of Mathematics

University of Kentucky

9 January 2019

Table of Contents

Preface vii

1 Expressions and Equations 1

1.1 Arithmetic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Answers for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Factorials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4


1.3 Factoring and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6


1.4 Algebraic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9


1.5 Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11


1.6 Basic Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13


2 Functions 15

2.1 Function Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16


2.2 Interpreting Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


2.3 Reading Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26


iv Table of Contents

2.4 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


2.5 Piecewise Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


2.6 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


2.7 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47


2.8 Logarithm Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


2.9 Exponential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.9.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51


3 Trigonometry 53

3.1 Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


3.2 The Unit Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58


3.3 Sine and Cosine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60


3.4 Tangent, Cosecant, Secant, and Cotangent . . . . . . . . . . . . . . . . . . . . . . . 63

3.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63


3.5 Right Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66


3.6 Trigonometric Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


3.7 The Five Fundamental Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73


Table of Contents v

4 Advanced Trigonometry and Algebra 774.1 Arcsine, Arccosine, and Arctangent . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.1.2 Answers for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Inverse Trigonometric Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.2 Answers for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3 Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3.2 Answers for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.2 Answers for Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

vi Table of Contents

Preface

Math textbooks can be very expensive and difficult to read. Many students have told me thatthe homework exercises were the only usable part of a textbook, for which they paid upwards of$100-$200. So I came up with a solution to help ease the financial burden. The goal of this bookletis to provide a comprehensive set of homework exercises with the required breadth and depth foran entire semester of an algebra and trigonometry course.

My hope is that this booklet will allow students to choose whatever resources are best for themwhile they take an algebra and trig class. You may still buy a standard textbook if needed. Butyou may also seek out less expensive/free alternatives, such as purchasing a Dover book, searchingonline for free material, or just using the notes provided during class lectures. This approach mayforce students to take a more proactive role in learning, which may not be a bad thing. At the veryleast, it allows students to avoid buying a potentially expensive and useless textbook.

This book is based on the work by Dr. Carl Stitz of Lakeland Community College and Dr. JeffZeager of Lorain County Community College. Editing for the University of Kentucky version wasdone by Andrew Cassidy and myself.

If you have any questions, comments, or concerns, feel free to contact me at [email protected].

Take THAT you expensive textbooks! I’m coming for you.

Jason Terry9 January 2019

viii Preface

Chapter 1

Expressions and Equations

2 Expressions and Equations

1.1 Arithmetic Expressions

1.1.1 Exercises

In Exercises 1 - 33, perform the indicated operations. Simplify like terms and reduce fractions tolowest terms.

1. 5− 2 + 3 2. 5− (2 + 3) 3.2

3− 4

7

4.3

8+

5

125.

5− 3

−2− 46.

2(−3)

3− (−3)

7.2(3)− (4− 1)

22 + 18.

4− 5.8

2− 2.19.

1− 2(−3)

5(−3) + 7

10.5(3)− 7

2(3)2 − 3(3)− 911.1

2[(−1)2 − 1]

[(−1)2 + 1]212.

(−2)2 − (−2)− 6

(−2)2 − 4

13.3− 4

9

−2− (−3)14.

23 −

45

4− 710

15.2(43

)1−

(43

)216.

1−(53

) (35

)1 +

(53

) (35

) 17.

(2

3

)−518. 3−1 − 4−2

19.1 + 2−3

3− 4−120.

3 · 5100

12 · 59821.√

32 + 42

22.√

12−√

75 23. (−8)2/3 − 9−3/2 24.(−32

9

)−3/525.

√(3− 4)2 + (5− 2)2 26.

√(2− (−1))2 +

(12 − 3

)227.

√(√

5− 2√

5)2 + (√

18−√

8)2 28.−12 +

√18

21

29.−2−

√(2)2 − 4(3)(−1)

2(3)30.−(−4) +

√(−4)2 − 4(1)(−1)

2(1)

31. 2(−5)(−5 + 1)−1 + (−5)2(−1)(−5 + 1)−2

32. 3√

2(4) + 1 + 3(4)(12

)(2(4) + 1)−1/2(2)

33. 2(−7) 3√

1− (−7) + (−7)2(13

)(1− (−7))−2/3(−1)

1Recall: there is no mathematical distinction between parentheses and brackets in computations; they’re just usedto organize groupings more clearly for your eyes.

1.1 Arithmetic Expressions 3

1.1.2 Answers for Exercises

1. 6 2. 0 3.2

214.

19

24

5. −1

36. −1 7.

3

58. 18

9. −7

810. Undefined. 11. 0 12. Undefined.

13.23

914. − 4

9915. −24

716. 0

17.243

3218.

13

4819.

9

2220.

25

4

21. 5 22. −3√

3 23.107

2724. −3 5

√3

8= −36/5

8

25.√

10 26.

√61

227.√

7

28.−4 +

√2

729. −1 30. 2 +

√5

31.15

1632. 13 33. −385

12


1.2 Factorials

1.2.1 Exercises

In Exercises 1 - 18, simplify the given expression to lowest terms. Assume all quantities are defined.

1. (3!)2 2.10!

7!3.

7!

233!

4. 3!4! 5.5!

3!6.

100!

98!2!

7.9!

4!3!2!8.

(n+ 1)!

n!9.

(k − 1)!

(k + 2)!

10.(2n+ 1)!

(2n− 1)!11.

6n+17nn!

6n7n+1(n+ 1)!12.

2k+13k(k + 3)!

2k−13k+1(k + 1)!

13.8!0!

3!5!14.

117!

117!0!15.

n!

(n− 2)!2!

16.(3n)!

[3(n+ 1)]!17.

2n−15n(2n− 1)!

2n5n−1(2n)!18.

[2(n+ 1)]!(3n)!

(2n)!(3n+ 1)!

1.2 Factorials 5


1. 36 2. 720 3. 105

4. 144 5. 20 6. 4950

7. 1260 8. n+ 1 9. 1k(k+1)(k+2)

10. 2n(2n+ 1) 11. 67(n+1) 12. 4(k+3)(k+2)

3

13. 56 14. 1 15. n(n−1)2

16. 1(3n+3)(3n+2)(3n+1) 17. 5

4n 18. (2n+2)(2n+1)(3n+1)


1.3 Factoring and Equations

1.3.1 Exercises

In Exercises 1 - 21, factor the given expression completely. (Hint: Check your answer by multipli-cation.)

1. 2x− 10x2 2. 12t5 − 8t3 3. 16xy2 − 12x2y

4. 5(m+ 3)2 − 4(m+ 3)3 5. (2x−1)(x+3)−4(2x−1) 6. t2(t− 5) + t− 5

7. w2 − 121 8. 49− 4t2 9. 81t4 − 16

10. 9z2 − 64y4 11. (y + 3)2 − 4y2 12. (x+ h)3 − (x+ h)

13. y2 − 24y + 144 14. 25t2 + 10t+ 1 15. 12x3 − 36x2 + 27x

16. x2 − 5x− 14 17. y2 − 12y + 27 18. 3t2 + 16t+ 5

19. 3m3 + 9m2 − 12m 20. m4 + 10m2 + 25 21. x3 − 5x2 − 9x+ 45

In Exercises 22 - 33, solve the given equation using factoring. Check your answers.

22. (7x+ 3)(x− 5) = 0 23. (2t− 1)2(t+ 4) = 0 24. (y2 + 4)(3y2 + y− 10) = 0

25. 4t = t2 26. y + 3 = 2y2 27. 26x = 8x2 + 21

28. 16x4 = 9x2 29. w(6w + 11) = 10 30. 2w2+5w+2 = −3(2w+1)

31. x2(x− 3) = 16(x− 3) 32. (2t+ 1)3 = (2t+ 1)33.

8t2

3= 2t+ 3

In Exercises 34 - 39, solve the given equation for the indicated variable. Simplify like terms andreduce to a single fraction to lowest terms.

34. Solve for y:1− 2y

y + 3= x 35. Solve for y: x = 3− 2

1− y

36.1Solve for T2:V1

T1

=V2

T2

37. Solve for t0:t0

1− t0t1= 2

38. Solve for R:1

R=

1

R1+

1

R239. Solve for vs: L =

1√1− vs

c2

1Recall: subscripts on variables have no mathematical meaning; they’re just used to distinguish variables. In otherwords, treat quantities like ‘V1’ and ‘V2’ as two different variables as you would ‘x’ and ‘y.’

1.3 Factoring and Equations 7

In Exercises 40 - 45, compute all real solutions.

40. (2x+ 1)3 + 8 = 0 41.(1− 2y)4

3= 27 42.

1

1 + 2t3= 4

43.√

3x+ 1 = 4 44. 5− 3√t2 + 1 = 1 45. x+ 1 =

√3x+ 7



1. 2x(1− 5x) 2. 4t3(3t2 − 2) 3. 4xy(4y − 3x)

4. −(m+ 3)2(4m+ 7) 5. (2x− 1)(x− 1) 6. (t− 5)(t2 + 1)

7. (w − 11)(w + 11) 8. (7− 2t)(7 + 2t) 9. (3t− 2)(3t+ 2)(9t2 + 4)

10. (3z − 8y2)(3z + 8y2) 11. −3(y − 3)(y + 1) 12. (x+h)(x+h−1)(x+h+1)

13. (y − 12)2 14. (5t+ 1)2 15. 3x(2x− 3)2

16. (x− 7)(x+ 2) 17. (y − 9)(y − 3) 18. (3t+ 1)(t+ 5)

19. 3m(m− 1)(m+ 4) 20. (m2 + 5)2 21. (x− 3)(x+ 3)(x− 5)

22. x = −3

7or x = 5 23. t =

1

2or t = −4 24. y =

5

3or y = −2

25. t = 0 or t = 4 26. y = −1 or y =3

227. x =

3

2or x =

7

4

28. x = 0 or x = ±3

429. w = −5

2or w =

2

330. w = −5 or w = −1

2

31. x = 3 or x = ±4 32. t = −1, t = −1

2, or t = 0 33. t = −3

4or t =

3

2

34. y =1− 3x

x+ 235. y =

x− 1

x− 3

36. T2 =V2T1

V1

37. t0 =2

2t1 + 1

38. R =R1R2

R1 +R239. vs =

c2(L2 − 1)

L2

40. x = −3

241. y = −1, 2 42. t = −

3√

3

2

43. x = 5 44. t = ±3√

7 45. x = 3

1.4 Algebraic Expressions 9

1.4 Algebraic Expressions

1.4.1 Exercises

In Exercises 1 - 18, perform the indicated operations and simplify into a single fraction in lowestterms using only positive exponents. Assume that all quantities are defined.

1.x2 − 9

x2· 3x

x2 − x− 62.

t2 − 2t

t2 + 1÷ (3t2 − 2t− 8) 3.

4y − y2

2y + 1÷ y2 − 16

2y2 − 5y − 3

4.x

3x− 1− 1− x

3x− 15.

2

w − 1− w2 + 1

w − 16.

2− y3y− 1− y

3y+y2 − 1

3y

7. b+1

b− 3− 2 8.

2x

x− 4− 1

2x+ 19.

m2

m2 − 4+

1

2−m

10.

2

x− 2

x− 111.

3

2− h− 3

2h

12.

1

x+ h− 1

xh

13. 3w−1 − (3w)−1 14. −2y−1 + 2(3− y)−2 15. 3(x− 2)−1 − 3x(x− 2)−2

16.t−1 + t−2

t−317.

2(3 + h)−2 − 2(3)−2

h18.

(7− x− h)−1 − (7− x)−1

h

In Exercises 19 - 27, simplify the given expression into lowest terms using only positive exponents.Assume that all quantities are defined.

19. (x−2x−3)4 20.2m−4

(2m−4)321.

(a3/2√bc2

)−2

22.

(√xy−2

yx−7/4

)4

23.(a−1 3√b · a−4/3b2

)224.

3 4√y

4x−2/3y3/2 · 3√y

25.(2pm−1q0)−4 · 2m−1p3

2pq226. (27x−3/2y5/2)−2/3 27.

(3a−2bc)(82/3a3b3c−3/2)

24ab−1/2√c



1.3(x+ 3)

x(x+ 2)2.

t

(3t+ 4)(t2 + 1)3. −y(y − 3)

y + 4

4.2x− 1

3x− 15. −w − 1 6.

y

3

7.b2 − 5b+ 7

b− 38.

4x2 + x+ 4

(x− 4)(2x+ 1)9.

m+ 1

m+ 2

10. −2

x11.

3

4− 2h12. − 1

x(x+ h)

13.8

3w14. −2(y2 − 7y + 9)

y(y − 3)215. − 6

(x− 2)2

16. t2 + t 17. − 2(h+ 6)

9(h+ 3)218.

1

(7− x)(7− x− h)

19.1

x20 20.m8

421.

bc4

a3

22.x9

y1223.

b14/3

a14/324.

x2/3

4y7/4

25.m3

16p2q226.

x

9y5/3 27.b9/2

2c

1.5 Intervals 11

1.5 Intervals

1.5.1 Exercises

1. Fill in the chart below. The first row has been completed for you.

Set of Real Numbers Interval Notation Region on the Real Number Line

{x : −1 ≤ x < 5} [−1, 5) −1 5

[0, 3)

2 7

{x : −5 < x ≤ 0}

(−3, 3)

5 7

{x : x ≤ 3}

(−∞, 9)

4

{x : x ≥ −3}

In Exercises 2 - 10, rewrite the set using interval notation.

2. {x : x 6= −1} 3. {x : x 6= −3, 4} 4. {x : x 6= 0, 2}

5. {x : x 6= 0, ±4} 6. {x : x < 3 andx ≥ 2} 7. {x : x ≤ −3 orx > 0}

8. {x : x ≤ 2 andx > 3} 9. {x : x > 2 orx = ±1} 10. {x : 3 < x < 13, x 6= 4}



1.

Set of Real Numbers Interval Notation Region on the Real Number Line

{x : −1 ≤ x < 5} [−1, 5) −1 5

{x : 0 ≤ x < 3} [0, 3)0 3

{x : 2 < x ≤ 7} (2, 7]2 7

{x : −5 < x ≤ 0} (−5, 0] −5 0

{x : −3 < x < 3} (−3, 3) −3 3

{x : 5 ≤ x ≤ 7} [5, 7]5 7

{x : x ≤ 3} (−∞, 3]3

{x : x < 9} (−∞, 9)9

{x : x > 4} (4,∞)4

{x : x ≥ −3} [−3,∞) −3

2. (−∞,−1) ∪ (−1,∞) 3. (−∞,−3) ∪ (−3, 4) ∪ (4,∞)

4. (−∞, 0) ∪ (0, 2) ∪ (2,∞) 5. (−∞,−4) ∪ (−4, 0) ∪ (0, 4) ∪ (4,∞)

6. [2, 3) 7. (−∞,−3] ∪ (0,∞)

8. ∅ 9. {−1} ∪ {1} ∪ (2,∞)

10. (3, 4) ∪ (4, 13)

1.6 Basic Graphs 13

1.6 Basic Graphs

1.6.1 Exercises

In Exercises 1 - 9, graph the given equation. Then state its domain and range.

1. y = 1 2. y = x 3. y = x2

4. y = x3 5. y =√x 6. y = 3

√x

7. y = |x| 8. y =1

x9. y =

1

x2



You may use technology such as Desmos to check your graphs.

1. Domain: (−∞,∞)Range: {1}

2. Domain: (−∞,∞)Range: (−∞,∞)

3. Domain: (−∞,∞)Range: [0,∞)

4. Domain: (−∞,∞)Range: (−∞,∞)

5. Domain: [0,∞)Range: [0,∞)

6. Domain: (−∞,∞)Range: (−∞,∞)

7. Domain: (−∞,∞)Range: [0,∞)

8. Domain: (−∞, 0)∪(0,∞)Range: (−∞, 0) ∪ (0,∞)

9. Domain: (−∞, 0)∪(0,∞)Range: (0,∞)

Chapter 2

Functions

16 Functions

2.1 Function Notation

2.1.1 Exercises

For the functions f described in Exercises 1 - 6, compute f(2). Then compute and simplify anexpression for f(x) that takes a real number x and performs the following three steps in the ordergiven:

1. (1) multiply by 2; (2) add 3; (3) divide by 4.

2. (1) add 3; (2) multiply by 2; (3) divide by 4.

3. (1) divide by 4; (2) add 3; (3) multiply by 2.

4. (1) multiply by 2; (2) add 3; (3) take the square root.

5. (1) add 3; (2) multiply by 2; (3) take the square root.

6. (1) add 3; (2) take the square root; (3) multiply by 2.

In Exercises 7 - 10, use the given function f to compute f(0) and solve f(x) = 0.

7. f(x) = 2x− 1 8. f(x) = 3− 25x

9. f(x) = 2x2 − 6 10. f(x) = x2 − x− 12

In Exercises 11 - 16, use the given function f to compute the following expressions and simplify byexpanding/combining like terms.

� f(3) � f(−1) � f(32

)� f(4x) � 4f(x) � f(−x)

� f(x− 4) � f(x)− 4 � f(x2)

11. f(x) = 2x+ 1 12. f(x) = 3− 4x

13. f(x) = 2− x2 14. f(x) = x2 − 3x+ 2

15. f(x) = 6 16. f(x) = 0

2.1 Function Notation 17

In Exercises 17 - 22, use the given function f to compute the following expressions and simplify byexpanding/combining like terms.

� f(2) � f(−2) � f(2a)

� 2f(a) � f(a+ 2) � f(a) + f(2)

� f

(2

a

)�f(a)

2� f(a+ h)

17. f(x) = 2x− 5 18. f(t) = 5− 2t

19. f(w) = 2w2 − 1 20. f(q) = 3q2 + 3q − 2

21. f(r) = 117 22. f(z) =z

2

In Exercises 23 - 28, use the given function to computef(x+ h)− f(x)

hand simplify by reducing

to lowest terms.

23. f(x) = 2x− 1 24. f(x) = −5

25. f(x) = 3− x2 26. f(x) = x3

27. f(x) =1

x28. f(x) =

1

x2

In Exercises 29 - 36, use the given pair of functions to compute the following expressions andsimplify by expanding/combining like terms.

� (g ◦ f)(x) � (f ◦ g)(t) � (f ◦ f)(x)

29. f(x) = 2x+ 3, g(t) = t2 − 9 30. f(x) = x2 − x+ 1, g(t) = 3t− 5

31. f(x) = 3x− 5, g(t) =√t 32. f(x) = |x+ 1|, g(t) =

√t

33. f(x) = 3− x2, g(t) =√t+ 1 34. f(x) = x2 − x− 1, g(t) =

√t− 5

35. f(x) = 3x− 1, g(t) =1

t+ 336. f(x) =

3x

x− 1, g(t) =

t

t− 3

18 Functions


1. f(2) = 74 , f(x) = 2x+3

4 2. f(2) = 52 , f(x) = 2(x+3)

4 = x+32

3. f(2) = 7, f(x) = 2(x4 + 3

)= 1

2x+ 6 4. f(2) =√

7, f(x) =√

2x+ 3

5. f(2) =√

10, f(x) =√

2(x+ 3) =√

2x+ 6 6. f(2) = 2√

5, f(x) = 2√x+ 3

7. For f(x) = 2x− 1, f(0) = −1 and f(x) = 0 when x = 12

8. For f(x) = 3− 25x, f(0) = 3 and f(x) = 0 when x = 15

2

9. For f(x) = 2x2 − 6, f(0) = −6 and f(x) = 0 when x = ±√

3

10. For f(x) = x2 − x− 12, f(0) = −12 and f(x) = 0 when x = −3 or x = 4

11. For f(x) = 2x+ 1

� f(3) = 7 � f(−1) = −1 � f(32

)= 4

� f(4x) = 8x+ 1 � 4f(x) = 8x+ 4 � f(−x) = −2x+ 1

� f(x− 4) = 2x− 7 � f(x)− 4 = 2x− 3 � f(x2)

= 2x2 + 1

12. For f(x) = 3− 4x

� f(3) = −9 � f(−1) = 7 � f(32

)= −3

� f(4x) = 3− 16x � 4f(x) = 12− 16x � f(−x) = 4x+ 3

� f(x− 4) = 19− 4x � f(x)− 4 = −4x− 1 � f(x2)

= 3− 4x2

13. For f(x) = 2− x2

� f(3) = −7 � f(−1) = 1 � f(32

)= −1

4

� f(4x) = 2− 16x2 � 4f(x) = 8− 4x2 � f(−x) = 2− x2

� f(x−4) = −x2+8x−14 � f(x)− 4 = −x2 − 2 � f(x2)

= 2− x4


14. For f(x) = x2 − 3x+ 2

� f(3) = 2 � f(−1) = 6 � f(32

)= −1

4

� f(4x) = 16x2 − 12x+ 2 � 4f(x) = 4x2 − 12x+ 8 � f(−x) = x2 + 3x+ 2

� f(x− 4) = x2− 11x+ 30 � f(x)− 4 = x2 − 3x− 2 � f(x2)

= x4 − 3x2 + 2

15. For f(x) = 6

� f(3) = 6 � f(−1) = 6 � f(32

)= 6

� f(4x) = 6 � 4f(x) = 24 � f(−x) = 6

� f(x− 4) = 6 � f(x)− 4 = 2 � f(x2)

= 6

16. For f(x) = 0

� f(3) = 0 � f(−1) = 0 � f(32

)= 0

� f(4x) = 0 � 4f(x) = 0 � f(−x) = 0

� f(x− 4) = 0 � f(x)− 4 = −4 � f(x2)

= 0

17. For f(x) = 2x− 5

� f(2) = −1 � f(−2) = −9 � f(2a) = 4a− 5

� 2f(a) = 4a− 10 � f(a+ 2) = 2a− 1 � f(a) + f(2) = 2a− 6

� f(2a

)= 4

a − 5= 4−5a

a

�f(a)2 = 2a−5

2� f(a+ h) = 2a+ 2h− 5

18. For f(x) = 5− 2x

� f(2) = 1 � f(−2) = 9 � f(2a) = 5− 4a

� 2f(a) = 10− 4a � f(a+ 2) = 1− 2a � f(a) + f(2) = 6− 2a

� f(2a

)= 5− 4

a= 5a−4

a

�f(a)2 = 5−2a

2� f(a+ h) = 5− 2a− 2h

20 Functions

19. For f(x) = 2x2 − 1

� f(2) = 7 � f(−2) = 7 � f(2a) = 8a2 − 1

� 2f(a) = 4a2 − 2 � f(a+ 2) = 2a2 + 8a+ 7 � f(a) + f(2) = 2a2 + 6

� f(2a

)= 8

a2− 1

= 8−a2a2

�f(a)2 = 2a2−1

2� f(a + h) = 2a2 + 4ah +

2h2 − 1

20. For f(x) = 3x2 + 3x− 2

� f(2) = 16 � f(−2) = 4 � f(2a) = 12a2 + 6a− 2

� 2f(a) = 6a2 + 6a− 4 � f(a+2) = 3a2+15a+16 � f(a)+f(2) = 3a2 +3a+14

� f(2a

)= 12

a2+ 6

a − 2

= 12+6a−2a2a2

�f(a)2 = 3a2+3a−2

2� f(a + h) = 3a2 + 6ah +

3h2 + 3a+ 3h− 2

21. For f(x) = 117

� f(2) = 117 � f(−2) = 117 � f(2a) = 117

� 2f(a) = 234 � f(a+ 2) = 117 � f(a) + f(2) = 234

� f(2a

)= 117 �

f(a)2 = 117

2� f(a+ h) = 117

22. For f(x) = x2

� f(2) = 1 � f(−2) = −1 � f(2a) = a

� 2f(a) = a � f(a+ 2) = a+22 � f(a) + f(2) = a

2 + 1= a+2

2

� f(2a

)= 1

a �f(a)2 = a

4� f(a+ h) = a+h

2

23. f(x+h)−f(x)h = 2 24. f(x+h)−f(x)

h = 0

25. f(x+h)−f(x)h = −2x− h 26. f(x+h)−f(x)

h = 3x2 + 3xh+ h2

27. f(x+h)−f(x)h = − 1

x(x+h) 28. f(x+h)−f(x)h = −2x−h

x2(x+h)2

29. For f(x) = 2x+ 3 and g(t) = t2 − 9

� (g ◦ f)(x) = 4x2 + 12x

� (f ◦ g)(t) = 2t2 − 15

� (f ◦ f)(x) = 4x+ 9


30. For f(x) = x2 − x+ 1 and g(t) = 3t− 5

� (g ◦ f)(x) = 3x2 − 3x− 2

� (f ◦ g)(t) = 9t2 − 33t+ 31

� (f ◦ f)(x) = x4 − 2x3 + 2x2 − x+ 1

31. For f(x) = 3x− 5 and g(t) =√t

� (g ◦ f)(x) =√

3x− 5

� (f ◦ g)(t) = 3√t− 5

� (f ◦ f)(x) = 9x− 20

32. For f(x) = |x+ 1| and g(t) =√t

� (g ◦ f)(x) =√|x+ 1|

� (f ◦ g)(t) = |√t+ 1| =

√t+ 1

� (f ◦ f)(x) = ||x+ 1|+ 1| = |x+ 1|+ 1

33. For f(x) = 3− x2 and g(t) =√t+ 1

� (g ◦ f)(x) =√

4− x2

� (f ◦ g)(t) = 2− t� (f ◦ f)(x) = −x4 + 6x2 − 6

34. For f(x) = x2 − x− 1 and g(t) =√t− 5

� (g ◦ f)(x) =√x2 − x− 6

� (f ◦ g)(t) = t− 6−√t− 5

� (f ◦ f)(x) = x4 − 2x3 − 2x2 + 3x+ 1

35. For f(x) = 3x− 1 and g(t) = 1t+3

� (g ◦ f)(x) = 13x+2

� (f ◦ g)(t) = − tt+3

� (f ◦ f)(x) = 9x− 4

36. For f(x) = 3xx−1 and g(t) = t

t−3

� (g ◦ f)(x) = x

� (f ◦ g)(t) = t

� (f ◦ f)(x) = 9x2x+1

22 Functions

2.2 Interpreting Functions

2.2.1 Exercises

1. The area enclosed by a square, in square inches, is a function of the length of one of its sides`, when measured in inches. This function is represented by the formula A(`) = `2 for ` > 0.Compute A(3) and solve A(`) = 36. Interpret your answers to each. Why is ` restricted to` > 0?

2. The area enclosed by a circle, in square meters, is a function of its radius r, when measuredin meters. This function is represented by the formula A(r) = πr2 for r > 0. Compute A(2)and solve A(r) = 16π. Interpret your answers to each. Why is r restricted to r > 0?

3. The volume enclosed by a cube, in cubic centimeters, is a function of the length of one of itssides s, when measured in centimeters. This function is represented by the formula V (s) = s3

for s > 0. Compute V (5) and solve V (s) = 27. Interpret your answers to each. Why is srestricted to s > 0?

4. The volume enclosed by a sphere, in cubic feet, is a function of the radius of the sphere r,when measured in feet. This function is represented by the formula V (r) = 4π

3 r3 for r > 0.

Compute V (3) and solve V (r) = 32π3 . Interpret your answers to each. Why is r restricted to

r > 0?

5. The height of an object dropped from the roof of an eight story building is modeled by thefunction: h(t) = −16t2 + 64, 0 ≤ t ≤ 2. Here, h(t) is the height of the object off the ground,in feet, t seconds after the object is dropped. Compute h(0) and solve h(t) = 0. Interpretyour answers to each. Why is t restricted to 0 ≤ t ≤ 2?

6. The temperature in degrees Fahrenheit t hours after 6 AM is given by T (t) = −12 t

2 + 8t+ 3for 0 ≤ t ≤ 12. Compute and interpret T (0), T (6) and T (12).

7. The function C(x) = x2 − 10x + 27 models the cost, in hundreds of dollars, to produce xthousand pens. Compute and interpret C(0), C(2) and C(5).

8. Suppose A(P ) gives the amount of money in a retirement account (in dollars) after 30 yearsas a function of the amount of the monthly payment (in dollars), P .

(a) What does A(50) mean?

(b) What is the significance of the solution to the equation A(P ) = 250000? .

(c) Explain what each of the following expressions mean: A(P +50), A(P )+50, and A(P )+A(50).

2.2 Interpreting Functions 23

9. Suppose P (t) gives the chance of precipitation (in percent) t hours after 8 AM.

(a) Write an expression which gives the chance of precipitation at noon.

(b) Write an equation you would solve which determines when the chance of precipitationis 50%.

In Exercises 10 - 15, compute the average rate of change of the function over the specified interval.

10. f(x) = x3, [−1, 2] 11. g(x) =1

x, [1, 5]

12. f(t) =√t, [0, 16] 13. g(t) = x2, [−3, 3]

14. F (s) =s+ 4

s− 3, [5, 7] 15. G(s) = 3s2 + 2s− 7, [−4, 2]

16. The height of an object dropped from the roof of a building is modeled by: h(t) = −16t2+64,for 0 ≤ t ≤ 2. Here, h(t) is the height of the object off the ground in feet t seconds after theobject is dropped. Compute and interpret the average rate of change of h over the interval[0, 2].

17. The temperature T (t) in degrees Fahrenheit t hours after 6 AM is given by:

T (t) = −1

2t2 + 8t+ 32, 0 ≤ t ≤ 12

(a) Compute and interpret T (4), T (8) and T (12).

(b) Compute and interpret the average rate of change of T over the interval [4, 8].

(c) Compute and interpret the average rate of change of T from t = 8 to t = 12.

(d) Compute and interpret the average rate of temperature change between 10 AM and 6PM.

18. Suppose C(x) = x2− 10x+ 27 represents the costs, in hundreds, to produce x thousand pens.Compute and interpret the average rate of change as production is increased from making3000 to 5000 pens.

24 Functions


1. A(3) = 9, so the area enclosed by a square with a side of length 3 inches is 9 square inches.The solutions to A(`) = 36 are ` = ±6. Since ` is restricted to ` > 0, we only keep ` = 6.This means for the area enclosed by the square to be 36 square inches, the length of the sideneeds to be 6 inches. Since ` represents a length, ` > 0.

2. A(2) = 4π, so the area enclosed by a circle with radius 2 meters is 4π square meters. Thesolutions to A(r) = 16π are r = ±4. Since r is restricted to r > 0, we only keep r = 4. Thismeans for the area enclosed by the circle to be 16π square meters, the radius needs to be 4meters. Since r represents a radius (length), r > 0.

3. V (5) = 125, so the volume enclosed by a cube with a side of length 5 centimeters is 125 cubiccentimeters. The solution to V (s) = 27 is s = 3. This means for the volume enclosed bythe cube to be 27 cubic centimeters, the length of the side needs to 3 centimeters. Since xrepresents a length, x > 0.

4. V (3) = 36π, so the volume enclosed by a sphere with radius 3 feet is 36π cubic feet. Thesolution to V (r) = 32π

3 is r = 2. This means for the volume enclosed by the sphere to be 32π3

cubic feet, the radius needs to 2 feet. Since r represents a radius (length), r > 0.

5. h(0) = 64, so at the moment the object is dropped off the building, the object is 64 feet off ofthe ground. The solutions to h(t) = 0 are t = ±2. Since we restrict 0 ≤ t ≤ 2, we only keept = 2. This means 2 seconds after the object is dropped off the building, it is 0 feet off theground. Said differently, the object hits the ground after 2 seconds. The restriction 0 ≤ t ≤ 2restricts the time to be between the moment the object is released and the moment it hitsthe ground.

6. T (0) = 3, so at 6 AM (0 hours after 6 AM), it is 3◦ Fahrenheit. T (6) = 33, so at noon (6hours after 6 AM), the temperature is 33◦ Fahrenheit. T (12) = 27, so at 6 PM (12 hoursafter 6 AM), it is 27◦ Fahrenheit.

7. C(0) = 27, so to make 0 pens, it costs1 $2700. C(2) = 11, so to make 2000 pens, it costs$1100. C(5) = 2, so to make 5000 pens, it costs $2000.

8. (a) The amount in the retirement account after 30 years if the monthly payment is $50.

(b) The solution to A(P ) = 250000 is what the monthly payment needs to be in order tohave $250,000 in the retirement account after 30 years.

(c) A(P + 50) is how much is in the retirement account in 30 years if $50 is added tothe monthly payment P . A(P ) + 50 represents the amount of money in the retirementaccount after 30 years if $P is invested each month plus an additional $50. A(P )+A(50)is the sum of money from two retirement accounts after 30 years: one with monthlypayment $P and one with monthly payment $50.

1This is called the ‘fixed’ or ‘start-up’ cost.

2.2 Interpreting Functions 25

9. (a) Since noon is 4 hours after 8 AM, P (4) gives the chance of precipitation at noon.

(b) We would need to solve P (t) = 0.5.

10. 23−(−1)32−(−1) = 3 11.

15− 1

15−1 = −1

512.

√16−√0

16−0 = 14 13. 32−(−3)2

3−(−3) = 0

14.7+47−3 −

5+45−3

7− 5= −7

815.

(3(2)2 + 2(2)− 7)− (3(−4)2 + 2(−4)− 7)

2− (−4)= −4

16. The average rate of change is h(2)−h(0)2−0 = −32. During the first two seconds after it is dropped,

the object has fallen at an average rate of 32 feet per second.

17. (a) T (4) = 56, so at 10 AM (4 hours after 6 AM), it is 56◦F. T (8) = 64, so at 2 PM (8 hoursafter 6 AM), it is 64◦F. T (12) = 56, so at 6 PM (12 hours after 6 AM), it is 56◦F.

(b) The average rate of change is T (8)−T (4)8−4 = 2. Between 10 AM and 2 PM, the temperature

increases, on average, at a rate of 2◦F per hour.

(c) The average rate of change is T (12)−T (8)12−8 = −2. Between 2 PM and 6 PM, the temperature

decreases, on average, at a rate of 2◦F per hour.

(d) The average rate of change is T (12)−T (4)12−4 = 0. Between 10 AM and 6 PM, the tempera-

ture, on average, remains constant.

18. The average rate of change is C(5)−C(3)5−3 = −2. As production is increased from 3000 to 5000

pens, the cost decreases at an average rate of $200 per 1000 pens produced (20¢ per pen.)

26 Functions

2.3 Reading Graphs

2.3.1 Exercises

In Exercises 1 - 4, determine whether or not the graph suggests y is a function of x. If so, statethe domain and range.

1.

x

y

−4 −3 −2 −1 1

−1

1

2

3

4

2.

x

y

−4 −3 −2 −1 1

−1

1

2

3

4

3.

x

y

−2 −1 1 2

1

2

3

4

5

4.

x

y

−3 −2 −1 1 2 3

−3

−2

−1

1

2

3

In Exercises 5 - 8, determine whether or not the graph suggests w is a function of v. If so, statethe domain and range.

5.

v

w

1 2 3 4 5 6 7 8 9

1

2

3

6.

v

w

−4 −3 −2 −1 1 2 3 4

1

2

3

4

2.3 Reading Graphs 27

7.

v

w

−4 −2 −1 1 2 4 5

−2

−1

1

8.

v

w

−5 −4 −3 −2 −1 1 2 3

−2

−1

1

2

3

4

In Exercises 9 - 12, determine whether or not the graph suggests T is a function of t. If so, statethe domain and range.

9.

t

T

−3−2−1 1 2 3

−5

−4

−3

−2

−1

1

2

3

4

5

6

7

8

9

10.

t

T

−5−4−3−2−1 1 2 3 4 5

11.

t

T

−5−4−3−2−1 1 2 3 4 5

−5

−4

−3

−2

−1

1

2

3

4

5

12.

t

T

−1 1 2 3 4 5 6

−5

−4

−3

−2

−1

1

2

3

4

5

28 Functions

In Exercises 13 - 16, determine whether or not the graph suggests H is a function of s. If so, statethe domain and range.

13.

s

H

−2 −1 1 2

1

2

3

4

14.

s

H

−2 −1 1 2

1

2

3

4

15.

s

H

−2 −1 1 2

1

2

3

4

16.

s

H

−2 −1 1 2

1

2

3

4

In Exercises 17 - 20, determine whether or not the graph suggests u is a function of t. If so, statethe domain and range.

17.

t

u

−2 −1 1 2

1

2

−2

−1

18.

t

u

−3 −2 −1 1 2 3

1

2

−2

−1

19.

t

u

−2 −1 1 2

1

2

−2

−1

20.

t

u

−2 −1 1 2

1

2

−2

−1


In Exercises 21 - 42, use the graph of y = f(x) given below to answer the question. You must useproper interval notation when appropriate.

x

y

(−5,−5)

(−4, 0)

(−3, 4)

(−2, 2)

(−1, 0)

(0,−1)

(1, 0)

(2, 3)

(3, 1)

−5 3 4 5

−5

−4

−3

−2

1

3

4

5

y = f(x)

21. Write the domain of f 22. Write the range of f

23. Write the maximum value, if it exists. 24. Write the minimum value, if it exists.

25. List the coordinates of the local maxi-mums, if any exist.

26. List the coordinates of the local minimums,if any exist.

27. Write the largest interval(s) where f is in-creasing.

28. Write the largest interval(s) where f is de-creasing.

29. Determine f(−2). 30. Solve f(x) = 4.

31. List the coordinates of the x-intercepts. 32. List the coordinates of the y-intercepts.

33. Solve f(x) = 0. 34. Solve f(x) ≥ 0.

35. Determine f(3). 36. Solve f(x) > 0.

37. Determine f(0). 38. Determine f(−5).

39. Solve f(x) ≤ 0. 40. Solve f(x) < 0.

41. How many solutions does f(x) = 1 have? 42. How many solutions does |f(x)| = 1 have?

30 Functions

In Exercises 43 - 64, use the graph of y = g(t) given below to answer the question. You must useproper interval notation when appropriate.

t

y

(−4, 0)

(−2,−5)

(0, 0)

(2, 3)

(2, 5)

(4, 0)−3 −2 −1

−5

−4

−3

−1

1

2

3

4

5

y = g(t)

43. Write the domain of g. 44. Write the range of g.

45. Write the maximum value, if it exists. 46. Write the minimum value, if it exists.

47. List the coordinates of the local maxi-mums, if any exist.

48. List the coordinates of the local minimums,if any exist.

49. Write the largest interval(s) where g is in-creasing.

50. Write the largest interval(s) where g is de-creasing.

51. Determine g(2). 52. Solve g(t) = −5.

53. List the coordinates of the t-intercepts. 54. List the coordinates of the y-intercepts.

55. Solve g(t) = 0. 56. Solve g(t) ≤ 0.

57. Determine g(−4). 58. Solve g(t) < 0.

59. Determine g(0). 60. Determine g(4).

61. Solve g(t) ≥ 0. 62. Solve g(t) > 0.

63. How many solutions does [g(t)]2 = 9 have? 64. How many solutions does |g(t)| = 5 have?



1. FunctionDomain: {−4, −3, −2, −1, 0, 1}Range: {−1, 0, 1, 2, 3, 4}

2. Not a function

3. FunctionDomain: (−∞,∞)Range: [1,∞)

4. Not a function

5. FunctionDomain: [2,∞)Range: [0,∞)

6. FunctionDomain: (−∞,∞)Range: (0, 4]

7. Not a function 8. FunctionDomain: [−5,−3) ∪ (−3, 3)Range: (−2,−1) ∪ [0, 4)

9. FunctionDomain: [−2,∞)Range: [−3,∞)

10. Not a function

11. FunctionDomain: (−5, 4)Range: (−4, 4)

12. FunctionDomain: [0, 3) ∪ (3, 6]Range: (−4,−1] ∪ [0, 4]

13. FunctionDomain: (−∞,∞)Range: (−∞, 4]

14. FunctionDomain: (−∞,∞)Range: (−∞, 4]

15. FunctionDomain: [−2,∞)Range: (−∞, 3]

16. FunctionDomain: (−∞,∞)Range: (−∞,∞)

17. FunctionDomain: (−∞, 0] ∪ (1,∞)Range: (−∞, 1] ∪ {2}

18. FunctionDomain: [−3, 3]Range: [−2, 2]

19. Not a function 20. FunctionDomain: (−∞,∞)Range: {2}

32 Functions

21. [−5, 3] 22. [−5, 4] 23. f(−3) = 4

24. f(−5) = −5 25. (−3, 4), (2, 3) 26. (0,−1)

27. [−5,−3] ∪ [0, 2] 28. [−3, 0] ∪ [2, 3] 29. f(−2) = 2

30. x = −3 31. (−4, 0), (−1, 0), (1, 0) 32. (0,−1)

33. x = −4,−1, 1 34. [−4,−1] ∪ [1, 3] 35. f(3) = 1

36. (−4,−1) ∪ (1, 3] 37. f(0) = −1 38. f(−5) = −5

39. [−5,−4] ∪ [−1, 1] 40. [−5, 4) ∪ (−1, 1) 41. Four.

42. Six. 43. [−4, 4] 44. [−5, 5)

45. None. 46. g(−2) = −5 47. None.

48. (−2,−5), (2, 3) 49. [−2, 2) 50. [−4,−2] ∪ (2, 4]

51. g(2) = 3 52. t = −2 53. (−4, 0), (0, 0), (4, 0)

54. (0, 0) 55. t = −4, 0, 4 56. [−4, 0] ∪ {4}

57. g(−4) = 0 58. (−4, 0) 59. g(0) = 0

60. g(4) = 0 61. [0, 4] 62. (0, 4)

63. Five. 64. One.

2.4 Transformations 33

2.4 Transformations

2.4.1 Exercises

In Exercises 1 - 9, use the graph of y = f(x) below to graph the given transformed function.

x

y

(−2, 2)

(0, 0)

(2, 2)

−4 −3 −2 −1 2 3 4

1

2

3

4

Graph of y = f(x) for Ex. 1 - 9

1. y = f(x) + 1 2. y = f(x)− 2 3. y = f(x+ 1)

4. y = f(x− 2) 5. y = 2f(x) 6. y = f(2x)

7. y = 2− f(x) 8. y = f(2− x) 9. y = 2− f(2− x)

In Exercises 10 - 18, use the graph of y = g(t) below to graph the given transformed function.

t

y

(−2, 0)

(0, 4)

(2, 0)

(4,−2)

−4 −3 −1 1 3 4

−4

−3

−2

−1

1

2

3

4

Graph of y = g(t) for Ex. 10 - 18

10. y = g(t)− 1 11. y = g(t+ 1) 12. y = 12g(t)

13. y = g(2t) 14. y = −g(t) 15. y = g(−t)

16. y = g(t+ 1)− 1 17. y = 1− g(t) 18. y = 12g(t+ 1)− 1

34 Functions

In Exercises 19 - 30, use the graph of y = f(x) below to graph the given transformed function.

(−3, 0)

(0, 3)

(3, 0)x

y

−3 −2 −1 1 2 3

−1

1

2

Graph of y = f(x) for Ex. 19 - 30

19. g(x) = f(x) + 3 20. h(x) = f(x)− 12 21. j(x) = f

(x− 2

3

)22. a(x) = f(x+ 4) 23. b(x) = f(x+ 1)− 1 24. c(x) = 3

5f(x)

25. d(x) = −2f(x) 26. k(x) = f(23x)

27. m(x) = −14f(3x)

28. n(x) = 4f(x− 3)− 6 29. p(x) = 4 + f(1− 2x) 30. q(x) = −12f(x+42

)− 3

In Exercises 31 - 42, sketch the graph of the given function using transformations. Start with thebasic of the function and graph/label each stage of its transformation.

31. f(x) = (x+ 2)2 − 1 32. f(x) = 3−√x+ 1 33. f(x) = −(x− 2)2 + 3

34. f(x) = 2√x− 3 35. f(x) =

1

2

√x− 3 36. f(x) =

√2x− 6

37. f(x) = 2− |x− 3| 38. f(x) = 2 + 3√

3− x 39. f(x) = −(x− 1)3 − 2

40. f(x) =1

x+ 2− 1 41. f(x) = 1− 2

x42. f(x) =

1

(x− 1)2+ 2



1. y = f(x) + 1

x

y

(−2, 3)

(0, 1)

(2, 3)

−4 −3 −2 −1 1 2 3 4

1

2

3

4

2. y = f(x)− 2

x

y

(−2, 2)

(0,−2)

(2, 2)−4 −1 1 4

−2

−1

1

2

3. y = f(x+ 1)

x

y

(−3, 2)

(−1, 0)

(1, 2)

−5 −4 −3 1 2 3

1

2

3

4

4. y = f(x− 2)

x

y

(0, 2)

(2, 0)

(4, 2)

−2 −1 3 4 5 6

1

2

4

5. y = 2f(x)

x

y

(−2, 4)

(0, 0)

(2, 4)

−4 −3 −2 −1 2 3 4

1

2

3

4

6. y = f(2x)

x

y

(−1, 2)

(0, 0)

(1, 2)

−4 −3 −2 −1 2 3 4

1

2

3

4

7. y = 2− f(x)

x

y

(−2, 0)

(0, 2)

(2, 0)−4 −3 3 4

1

2

8. y = f(2− x)

x

y

(0, 2)

(2, 0)

(4, 2)

−2 −1 3 4 5 6

1

2

4

36 Functions

9. y = 2− f(2− x)

x

y

(0, 0)

(2, 2)

(4, 0)−2 −1 2 5 6

1

2

3

4

10. y = g(t)− 1

t

y

(−2,−1)

(0, 3)

(2,−1)

(4,−3)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4

11. y = g(t+ 1)

t

y

(−3, 0)

(−1, 4)

(1, 0)

(3,−2)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4

12. y = 12g(t)

t

y

(−2, 0)

(0, 2)

(2, 0) (4,−1)

−4 −3 −1 1 3 4

−4

−3

−2

−1

1

2

3

4

13. y = g(2t)

t

y

(−1, 0)

(0, 4)

(1, 0)

(2,−2)

−4 −3 −2 2 3 4

−4

−3

−2

1

2

3

4

14. y = −g(t)

t

y

(−2, 0)

(0,−4)

(2, 0)

(4, 2)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4


15. y = g(−t)

t

y

(2, 0)

(0, 4)

(−2, 0)

(−4,−2)

−4 −3 −1 1 3 4

−4

−3

−2

−1

1

2

3

4

16. y = g(t+ 1)− 1

t

y

(−3,−1)

(−1, 3)

(1,−1)

(3,−3)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4

17. y = 1− g(t)

t

y

(−2, 1)

(0,−3)

(2, 1)

(4, 3)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4

18. y = 12g(t+ 1)− 1

t

y

(−3,−1)

(−1, 1)

(1,−1)

(3,−2)

−4 −3 −1−2 1 2 3 4

−4

−3

−2

−1

1

2

3

4

19. g(x) = f(x) + 3

(−3, 3)

(0, 6)

(3, 3)

x

y

−3 −2 −1 1 2 3

−1

1

2

3

4

5

6

20. h(x) = f(x)− 12

(−3,− 1

2

)

(0, 5

2

)

(3,− 1

2

)x

y

−3 −2 −1 1 2

−1

1

2

3

38 Functions

21. j(x) = f(x− 2

3

)

(− 7

3, 0

)

(23, 3

)

(113

, 0)x

y

−3 −2 −1 1 2 3

−1

1

2

3

22. a(x) = f(x+ 4)

(−7, 0)

(−4, 3)

(−1, 0)x

y

−7 −6 −5 −4 −3 −2 −1

1

2

3

23. b(x) = f(x+ 1)− 1

(−4,−1)

(−1, 2)

(2,−1)

x

y

−4 −3 −2 −1 1 2

−1

1

2

24. c(x) = 35f(x)

(−3, 0)

(0, 9

5

)

(3, 0)x

y

−3 −2 −1 1 2 3

−1

1

2

25. d(x) = −2f(x)

(−3, 0)

(0,−6)

(3, 0)

x

y

−3 −2 −1 1 2 3

−6

−5

−4

−3

−2

−1

26. k(x) = f(23x)

(− 9

2, 0

)

(0, 3)

(92, 0

)x

y

−4 −3 −2 −1 1 2 3 4

−1

1

2

3


27. m(x) = −14f(3x)

(−1, 0)

(0,− 3

4

)(1, 0)

x

y

−1 1

−1

28. n(x) = 4f(x− 3)− 6

(0,−6)

(3, 6)

(6,−6)

x

y

1 2 3 4 5 6

−6

−5

−4

−3

−2

−1

1

2

3

4

5

6

29. p(x) = 4 + f(1− 2x) = f(−2x+ 1) + 4

(−1, 4)

(12, 7

)

(2, 4)

x

y

−1 1 2

−1

1

2

3

4

5

6

7

30. q(x) = − 12f(x+42

)− 3 = − 1

2f(12x+ 2

)− 3

(−10,−3)

(−4,− 9

2

)(2,−3)

x

y

−10−9−8−7−6−5−4−3−2−1 1 2

−4

−3

−2

−1

In Exercises 31 - 42, you may use technology such as Desmos to check your graphs.

40 Functions

2.5 Piecewise Functions

2.5.1 Exercises

In Exercises 1 - 4, graph the function. State the domain, range, and axis intercepts, if any.

1. f(x) =

{4− x if x ≤ 3

2 if x > 32. g(x) =

{2− x if x < 2

x− 2 if x ≥ 2

3. F (t) =

{−2t− 4 if t < 0

3t if t ≥ 0 4. G(t) =

−3 if t < 0

2t− 3 if 0 ≤ t ≤ 3

3 if t > 3

5. For n copies of the book Me and my Sasquatch, a print on-demand company charges C(n)dollars, where C(n) is determined by the formula

C(n) =

15n if 1 ≤ n ≤ 25

13.50n if 25 < n ≤ 50

12n if n > 50

(a) Compute and interpret C(20).

(b) How much does it cost to order 50 copies of the book? What about 51 copies?

(c) Your answer to 5b should get you thinking. Suppose a bookstore estimates it will sell50 copies of the book. How many books can, in fact, be ordered for the same price asthose 50 copies? (Round your answer to a whole number of books.)

6. An on-line comic book retailer charges shipping costs according to the following formula

S(n) =

{1.5n+ 2.5 if 1 ≤ n ≤ 14

0 if n ≥ 15

where n is the number of comic books purchased and S(n) is the shipping cost in dollars.

(a) What is the cost to ship 10 comic books?

(b) What is the significance of the formula S(n) = 0 for n ≥ 15?

7. The cost in dollars C(m) to talk m minutes a month on a mobile phone plan is modeled by

C(m) =

{25 if 0 ≤ m ≤ 1000

25 + 0.1(m− 1000) if m > 1000

(a) How much does it cost to talk 750 minutes per month with this plan?

(b) How much does it cost to talk 20 hours a month with this plan?

(c) Explain the terms of the plan verbally.

2.5 Piecewise Functions 41


1.

Domain: (−∞,∞)

Range: [1,∞)

y-intercept: (0, 4)

x-intercept: None

y

x

1

2

3

4

5

−1 1 2 3 4 5 6 7

2.


Range: [0,∞)

y-intercept: (0, 2)

x-intercept: (2, 0)

y

x

1

2

3

4

5

−1 1 2 3 4 5 6 7

3.


Range: (−4,∞)

y-intercept: (0, 0)

t-intercepts: (−2, 0), (0, 0)t

y

−2 −1 1

−4

−3

−2

−1

1

2

3

4.


Range: [−3, 3]

y-intercept: (0,−3)

t-intercept:(32 , 0)

= (1.5, 0)

y

t

−2

−1

1

2

3

−4−3−2−1 1 2 3 4

42 Functions

5. (a) C(20) = 300. It costs $300 for 20 copies of the book.

(b) C(50) = 675, $675. C(51) = 612, $612.

(c) 56 books.

6. (a) S(10) = 17.5, $17.50.

(b) There is free shipping on orders of 15 or more comic books.

7. (a) C(750) = 25, $25.

(b) C(1200) = 45, $45.

(c) It costs $25 for up to 1000 minutes and 10 cents per minute for each minute over 1000minutes.

2.6 Inverse Functions 43

2.6 Inverse Functions

2.6.1 Exercises

In Exercises 1 - 20, compute the inverse of the given function (assuming it is one-to-one).

1. f(x) = 6x− 2 2. f(x) = 42− x

3. g(t) =t− 2

3+ 4 4. g(t) = 1− 4 + 3t

5

5. f(x) =√

3x− 1 + 5 6. f(x) = 2−√x− 5

7. g(t) = 3√t− 1− 4 8. g(t) = 1− 2

√2t+ 5

9. f(x) = 5√

3x− 1 10. f(x) = 3− 3√x− 2

11. g(t) = t2 − 10t, t ≥ 5 12. g(t) = 3(t+ 4)2 − 5, t ≤ −4

13. f(x) = x2 − 6x+ 5, x ≤ 3 14. f(x) = 4x2 + 4x+ 1, x < −1

15. g(t) =3

4− t16. g(t) =

t

1− 3t

17. f(x) =2x− 1

3x+ 418. f(x) =

4x+ 2

3x− 6

19. g(t) =−3t− 2

t+ 320. g(t) =

t− 2

2t− 1

In Exercises 21 - 24, each graph represents a one-to-one function. Graph its inverse and identifythe asymptote(s) if any are present.

21. y = f(x)

x

y

(0, 1)

(1, 2)

(2, 4)

−2 −1 1 2

2

3

4

5

Asymptote: y = 0.

22. y = g(t)

t

y

(1, 0)

(0, 2)

(−2, 4)

−3 −2 −1 2

−2−3−4−5

1

3

54

Asymptote: t = 2.

44 Functions

23. y = S(t)

t

y

(−4,−3)

(0, 0)

(4, 3)

−4 −3 −2 −1 2 3 4

−3

−2

−1

1

2

3

Domain: [−4, 4].

24. y = R(s)

s

y

(− 1

2,− 3

2

)(0, 0)

(12, 32

)

Asymptotes: y = ±3.

2.6 Inverse Functions 45


1. f−1(x) =x+ 2

62. f−1(x) = 42− x

3. g−1(t) = 3t− 10 4. g−1(t) = −53 t+ 1

3

5. f−1(x) = 13(x− 5)2 + 1

3 , x ≥ 5 6. f−1(x) = (x− 2)2 + 5, x ≤ 2

7. g−1(t) = 19(t+ 4)2 + 1, t ≥ −4 8. g−1(t) = 1

8(t− 1)2 − 52 , t ≤ 1

9. f−1(x) = 13x

5 + 13 10. f−1(x) = −(x− 3)3 + 2

11. g−1(t) = 5 +√t+ 25 12. g−1(t) = −

√t+53 − 4

13. f−1(x) = 3−√x+ 4 14. f−1(x) = −

√x+12 , x > 1

15. g−1(t) =4t− 3

t16. g−1(t) =

t

3t+ 1

17. f−1(x) =4x+ 1

2− 3x18. f−1(x) =

6x+ 2

3x− 4

19. g−1(t) =−3t− 2

t+ 320. g−1(t) =

t− 2

2t− 1

21. y = f−1(x). Asymptote: x = 0.

x

y

(1, 0)

(2, 1)

(4, 2)

−2

−1

1

2

3 4 5

22. y = g−1(t). Asymptote: y = 2.

t

y

(2, 0)

(0, 1)

(4,−2)

−3

−2

−1

2

−2−3−4−5 1 3 54

46 Functions

23. y = S−1(t). Domain [−3, 3].

t

y

(−3,−4)

(0, 0)

(3, 4)

−4

−3

−2

−1

2

3

4

1

−3 −2 −1 2 3

24. y = R−1(s). Asymptotes: s = ±3.

s

y

(− 3

2,− 1

2

)(0, 0)

(32, 12

)

2.7 Logarithms 47

2.7 Logarithms

2.7.1 Exercises

In Exercises 1 - 15, rewrite the exponential equations as logarithmic equations and rewrite thelogarithmic equations as exponential equations.

1. 23 = 8 2. 5−3 = 1125 3. 45/2 = 32

4.(13

)−2= 9 5.

(425

)−1/2= 5

26. 10−3 = 0.001

7. e0 = 1 8. log5(25) = 2 9. log25(5) = 12

10. log3(

181

)= −4 11. log 4

3

(34

)= −1 12. log(100) = 2

13. log(0.1) = −1 14. ln(e) = 1 15. ln(

1√e

)= −1

2

In Exercises 16 - 42, evaluate the expression without using a calculator.

16. log3(27) 17. log6(216) 18. log2(32)

19. log6(

136

)20. log8(4) 21. log36(216)

22. log 15(625) 23. log 1

6(216) 24. log36(36)

25. log(

11000000

)26. log(0.01) 27. ln

(e3)

28. log4(8) 29. log6(1) 30. log13(√

13)

31. log36(

4√

36)

32. 7log7(3) 33. 36log36(216)

34. log36(36216

)35. ln

(e5)

36. log(

9√

1011)

37. log(

3√

105)

38. ln(

1√e

)39. log5

(3log3(5)

)40. log

(eln(100)

)41. log2

(3− log3(2)

)42. ln

(426 log(1)

)In Exercises 43 - 46, compute the domain of the function.

43. f(x) = ln(x2 + 1) 44. f(x) = log7(4x+ 8)

45. g(t) = ln(4t− 20) 46. g(t) = log(t2 + 9t+ 18

)

48 Functions


1. log2(8) = 3 2. log5(

1125

)= −3 3. log4(32) = 5

2

4. log 13(9) = −2 5. log 4

25

(52

)= −1

2 6. log(0.001) = −3

7. ln(1) = 0 8. 52 = 25 9. (25)12 = 5

10. 3−4 = 181 11.

(43

)−1= 3

412. 102 = 100

13. 10−1 = 0.1 14. e1 = e 15. e−12 = 1√

e

16. log3(27) = 3 17. log6(216) = 3 18. log2(32) = 5

19. log6(

136

)= −2 20. log8(4) = 2

3 21. log36(216) = 32

22. log 15(625) = −4 23. log 1

6(216) = −3 24. log36(36) = 1

25. log 11000000 = −6 26. log(0.01) = −2 27. ln

(e3)

= 3

28. log4(8) = 32 29. log6(1) = 0 30. log13

(√13)

= 12

31. log36(

4√

36)

= 14 32. 7log7(3) = 3 33. 36log36(216) = 216

34. log36(36216

)= 216 35. ln(e5) = 5 36. log

(9√

1011)

= 119

37. log(

3√

105)

= 53 38. ln

(1√e

)= −1

239. log5

(3log3 5

)= 1

40. log(eln(100)

)= 2 41. log2

(3− log3(2)

)= −1 42. ln

(426 log(1)

)= 0

43. (−∞,∞) 44. (−2,∞) 45. (5,∞)

46. (−∞,−6) ∪ (−3,∞)

2.8 Logarithm Rules 49

2.8 Logarithm Rules

2.8.1 Exercises

In Exercises 1 - 15, expand the given expression as a sum/difference of multiples of logarithms andsimplify. Assume that all quantities are defined.

1. ln(x3y2) 2. log2

(128

x2 + 4

)3. log5

[( z25

)3]

4. log(1.23× 1037) 5. ln

(√z

xy

)6. log5

(x2 − 25

)7. log√2

(4x3)

8. log 13(9x(y2 − 4)) 9. log

(1000x3y5

)10. log3

(x2

81y4

)11. ln

(4

√xy

ez

)12. log6

[(216

x3y

)4]

13. log

(100x

√y

3√

10

)14. log 1

2

(4

3√x2

y√z

)15. ln

(3√x

10√yz

)In Exercises 16 - 26, simplify the expression into a single logarithm. Assume that all quantities aredefined.

16. 4 ln(x) + 2 ln(y) 17. log2(x) + log2(y)− log2(z)

18. log3(x)− 2 log3(y) 19. 12 log3(x)− 2 log3(y)− log3(z)

20. 2 ln(x)− 3 ln(y)− 4 ln(z) 21. log(x)− 13 log(z) + 1

2 log(y)

22. −13 ln(x)− 1

3 ln(y) + 13 ln(z) 23. log5(x)− 3

24. 3− log(x) 25. log7(x) + log7(x− 3)− 2

26. ln(x) + 12

50 Functions


1. 3 ln(x) + 2 ln(y) 2. 7− log2(x2 + 4)

3. 3 log5(z)− 6 4. log(1.23) + 37

5. 12 ln(z)− ln(x)− ln(y) 6. log5(x− 5) + log5(x+ 5)

7. 3 log√2(x) + 4 8. −2 + log 13(x) + log 1

3(y + 2) + log 1

3(y − 2)

9. 3 + 3 log(x) + 5 log(y) 10. 2 log3(x)− 4− 4 log3(y)

11. 14 ln(x) + 1

4 ln(y)− 14 −

14 ln(z) 12. 12− 12 log6(x)− 4 log6(y)

13. 53 + log(x) + 1

2 log(y) 14. −2 + 23 log 1

2(x)− log 1

2(y)− 1

2 log 12(z)

15. 13 ln(x)− ln(10)− 1

2 ln(y)− 12 ln(z) 16. ln(x4y2)

17. log2(xyz

)18. log3

(xy2

)19. log3

(√x

y2z

)20. ln

(x2

y3z4

)21. log

(x√y

3√z

)22. ln

(3

√zxy

)23. log5

(x125

)24. log

(1000x

)25. log7

(x(x−3)

49

)26. ln (x

√e)

2.9 Exponential Equations 51

2.9 Exponential Equations

2.9.1 Exercises

In Exercises 1 - 30, solve the equation.

1. 24x = 8 2. 3(x−1) = 27 3. 52x−1 = 125

4. 42t = 12 5. 8t = 1

128 6. 2(t3−t) = 1

7. 37x = 814−2x 8. 9 · 37x =(19

)2x9. 32x = 5

10. 5−t = 2 11. 5t = −2 12. 3(t−1) = 29

13. (1.005)12x = 3 14. e−5730k = 12 15. 2000e0.1t = 4000

16. 500(1− e2t

)= 250 17. 70 + 90e−0.1t = 75 18. 30− 6e−0.1t = 20

19.100ex

ex + 2= 50 20.

5000

1 + 2e−3t= 2500 21.

150

1 + 29e−0.8t= 75

22. 25(45

)x= 10 23. e2x = 2ex 24. 7e2t = 28e−6t

25. 3(x−1) = 2x 26. 3(x−1) =(12

)(x+5) 27. 73+7x = 34−2x

28. e2t − 3et − 10 = 0 29. e2t = et + 6 30. 4t + 2t = 12

52 Functions


1. x = 34 2. x = 4 3. x = 2

4. t = −14 5. t = −7

3 6. t = −1, 0, 1

7. x = 1615 8. x = − 2

11 9. x =ln(5)

2 ln(3)

10. t = − ln(2)

ln(5)11. No solution. 12. t =

ln(29) + ln(3)

ln(3)

13. x =ln(3)

12 ln(1.005)14. k =

ln(12

)−5730

=ln(2)

573015. t =

ln(2)

0.1= 10 ln(2)

16. t = 12 ln

(12

)= −1

2 ln(2) 17. t =ln(

118

)−0.1

= 10 ln(18)

18. t = −10 ln(53

)= 10 ln

(35

)19. x = ln(2)

20. t = 13 ln(2) 21. t =

ln(

129

)−0.8

=5

4ln(29)

22. x =ln(25

)ln(45

) =ln(2)− ln(5)

ln(4)− ln(5)23. x = ln(2)

24. t = −18 ln

(14

)= 1

4 ln(2) 25. x =ln(3)

ln(3)− ln(2)

26. x =ln(3) + 5 ln

(12

)ln(3)− ln

(12

) =ln(3)− 5 ln(2)

ln(3) + ln(2)27. x =

4 ln(3)− 3 ln(7)

7 ln(7) + 2 ln(3)

28. t = ln(5) 29. t = ln(3) 30. t =ln(3)

ln(2)

Chapter 3

Trigonometry

54 Trigonometry

3.1 Angles

3.1.1 Exercises

In Exercises 1 - 16, graph the given angle in standard position, state the quadrant in which it lies,and then compute two coterminal angles, one of which is positive and the other negative.

1.π

3 2.5π

63. −11π

34.

5π

4

5.3π

46. −π

37.

7π

28.

π

4

9. −π2

10.7π

611. −5π

312. 3π

13. −2π 14. −π4

15.15π

416. −13π

6

In Exercises 17 - 24, convert the angle from degree measure into radian measure, giving the exactvalue in terms of π and reduced to lowest terms.

17. 0◦ 18. 240◦ 19. 135◦ 20. −270◦

21. −315◦ 22. 150◦ 23. 45◦ 24. −225◦

In Exercises 25 - 32, convert the angle from radian measure into degree measure.

25. π 26. −2π

327.

7π

628.

11π

6

29.π

330.

5π

331. −π

632.

π

2

3.1 Angles 55


1.π

3is a Quadrant I angle

coterminal with7π

3and −5π

3

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

2.5π

6is a Quadrant II angle

coterminal with17π

6and −7π

6

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

3. −11π


coterminal withπ

3and −5π

3

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

4.5π

4is a Quadrant III angle

coterminal with13π

4and −3π

4

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

5.3π

4is a Quadrant II angle

coterminal with11π

4and −5π

4

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

6. −π3

is a Quadrant IV angle

coterminal with5π

3and −7π

3

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

56 Trigonometry

7.7π

2lies on the negative y-axis

coterminal with3π

2and −π

2

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

8.π


coterminal with9π

4and −7π

4

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

9. −π2

lies on the negative y-axis

coterminal with3π

2and −5π

2

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

10.7π

6is a Quadrant III angle

coterminal with19π

6and −5π

6

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

11. −5π


coterminal withπ

3and −11π

3

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

12. 3π lies on the negative x-axis

coterminal with π and −π

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

3.1 Angles 57

13. −2π lies on the positive x-axis

coterminal with 2π and −4π

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

14. −π4

is a Quadrant IV angle

coterminal with7π

4and −9π

4

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

15.15π

4is a Quadrant IV angle

coterminal with7π

4and −π

4

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

16. −13π

6is a Quadrant IV angle

coterminal with11π

6and −π

6

x

y

−4−3−2−1 1 2 3 4−1−2−3−4

1

2

3

4

17. 0 18.4π

319.

3π

420. −3π

2

21. −7π

422.

5π

623.

π

424. −5π

4

25. 180◦ 26. −120◦ 27. 210◦ 28. 330◦

29. 60◦ 30. 300◦ 31. −30◦ 32. 90◦

58 Trigonometry

3.2 The Unit Circle

3.2.1 Exercises

Write the coordinates of all the points indicated on the unit circle below.

x

y

0, 2π

π

2

π

3π

2

π

4

π

6

π

33π

4

5π

6

2π

3

5π

4

7π

6

4π

3

7π

4

11π

6

5π

3

Important Points on the Unit Circle

3.2 The Unit Circle 59


x

y

(0, 1)

(1, 0)

(0,−1)

(−1, 0)

(√22 ,√22

)(√

32 ,

12

)

(12 ,√32

)(−√22 ,√22

)(−√32 ,

12

)

(−1

2 ,√32

)

(√22 ,−

√22

)(√

32 ,−

12

)

(12 ,−

√32

)(−√22 ,−

√22

)(−√32 ,−

12

)

(−1

2 ,−√32

)

0, 2π

π

2

π

3π

2

π

4

π

6

π

33π

4

5π

6

2π

3

5π

4

7π

6

4π

3

7π

4

11π

6

5π

3

Important Points on the Unit Circle

60 Trigonometry

3.3 Sine and Cosine

3.3.1 Exercises

In Exercises 1 - 20, determine the exact value of the cosine and sine of the given angle.

1. θ = 0 2. θ =π

43. θ =

π

34. θ =

π

2

5. θ =2π

36. θ =

3π

47. θ = π 8. θ =

7π

6

9. θ =5π

410. θ =

4π

311. θ =

3π

212. θ =

5π

3

13. θ =7π

414. θ =

23π

615. θ = −13π

216. θ = −43π

6

17. θ = −3π

418. θ = −π

619. θ =

10π

320. θ = 117π

In Exercises 21 - 29, determine all of the angles which satisfy the given equation.

21. sin(θ) =1

2 22. cos(θ) = −√

3

223. sin(θ) = 0

24. cos(θ) =

√2

225. sin(θ) =

√3

226. cos(θ) = −1

27. sin(θ) = −1 28. cos(θ) =

√3

229. cos(θ) = −1.001

In Exercises 30 - 38, determine all of the angles which satisfy the given equation.

30. cos(t) = 0 31. sin(t) = −√

2

232. cos(t) = 3

33. sin(t) = −1

234. cos(t) =

1

235. sin(t) = −2

36. cos(t) = 1 37. sin(t) = 1 38. cos(t) = −√

2

2

3.3 Sine and Cosine 61


1. cos(0) = 1, sin(0) = 0 2. cos(π

4

)=

√2

2, sin

(π4

)=

√2

2

3. cos(π

3

)=

1

2, sin

(π3

)=

√3

24. cos

(π2

)= 0, sin

(π2

)= 1

5. cos

(2π

3

)= −1

2, sin

(2π

3

)=

√3

26. cos

(3π

4

)= −√

2

2, sin

(3π

4

)=

√2

2

7. cos(π) = −1, sin(π) = 0 8. cos

(7π

6

)= −√

3

2, sin

(7π

6

)= −1

2

9. cos

(5π

4

)= −√

2

2, sin

(5π

4

)= −√

2

210. cos

(4π

3

)= −1

2, sin

(4π

3

)= −√

3

2

11. cos

(3π

2

)= 0, sin

(3π

2

)= −1 12. cos

(5π

3

)=

1

2, sin

(5π

3

)= −√

3

2

13. cos

(7π

4

)=

√2

2, sin

(7π

4

)= −√

2

214. cos

(23π

6

)=

√3

2, sin

(23π

6

)= −1

2

15. cos

(−13π

2

)= 0, sin

(−13π

2

)= −1 16. cos

(−43π

6

)= −√

3

2, sin

(−43π

6

)=

1

2

17. cos

(−3π

4

)= −√

2

2, sin

(−3π

4

)= −√

2

218. cos

(−π

6

)=

√3

2, sin

(−π

6

)= −1

2

19. cos

(10π

3

)= −1

2, sin

(10π

3

)= −√

3

220. cos(117π) = −1, sin(117π) = 0

21. sin(θ) =1

2when θ =

π

6+ 2πk or θ =

5π

6+ 2πk for any integer k.

22. cos(θ) = −√

3

2when θ =

5π

6+ 2πk or θ =

7π


23. sin(θ) = 0 when θ = πk for any integer k.

24. cos(θ) =

√2

2when θ =

π

4+ 2πk or θ =

7π


25. sin(θ) =

√3

2when θ =

π

3+ 2πk or θ =

2π


26. cos(θ) = −1 when θ = (2k + 1)π for any integer k.

27. sin(θ) = −1 when θ =3π


62 Trigonometry

28. cos(θ) =

√3

2when θ =

π

6+ 2πk or θ =

11π


29. cos(θ) = −1.001 has no solution.

30. cos(t) = 0 when t =π

2+ πk for any integer k.

31. sin(t) = −√

2

2when t =

5π

4+ 2πk or t =

7π


32. cos(t) = 3 has no solution.

33. sin(t) = −1

2when t =

7π

6+ 2πk or t =

11π


34. cos(t) =1

2when t =

π

3+ 2πk or t =

5π


35. sin(t) = −2 has no solution.

36. cos(t) = 1 when t = 2πk for any integer k.

37. sin(t) = 1 when t =π


38. cos(t) = −√

2

2when t =

3π

4+ 2πk or t =

5π


3.4 Tangent, Cosecant, Secant, and Cotangent 63

3.4 Tangent, Cosecant, Secant, and Cotangent

3.4.1 Exercises

In Exercises 1 - 20, determine the exact value or state that it is undefined.

1. tan(π

4

)2. sec

(π6

)3. csc

(5π

6

)4. cot

(4π

3

)5. tan

(−11π

6

)6. sec

(−3π

2

)7. csc

(−π

3

)8. cot

(13π

2

)9. tan (117π) 10. sec

(−5π

3

)11. csc (3π) 12. cot (−5π)

13. tan

(31π

2

)14. sec

(π4

)15. csc

(−7π

4

)16. cot

(7π

6

)17. tan

(2π

3

)18. sec (−7π) 19. csc

(π2

)20. cot

(3π

4

)In Exercises 21 - 24, use the given information to determine the quadrant in which the given anglelies.

21. sin(θ) > 0 and tan(θ) < 0. 22. cot(θ) > 0 and cos(θ) < 0.

23. sin(θ) > 0 and tan(θ) > 0. 24. cos(θ) > 0 and cot(θ) < 0.

In Exercises 25 - 39, determine all of the angles which satisfy the equation.

25. tan(θ) =√

3 26. sec(θ) = 2 27. csc(θ) = −1 28. cot(θ) =

√3

3

29. tan(θ) = 0 30. sec(θ) = 1 31. csc(θ) = 2 32. cot(θ) = 0

33. tan(θ) = −1 34. sec(θ) = 0 35. csc(θ) = −1

236. sec(θ) = −1

37. tan(θ) = −√

3 38. csc(θ) = −2 39. cot(θ) = −1

In Exercises 40 - 47, determine all of the angles which satisfy the equation.

40. cot(t) = 1 41. tan(t) =

√3

342. sec(t) = −2

√3

343. csc(t) = 0

44. cot(t) = −√

3 45. tan(t) = −√

3

346. sec(t) =

2√

3

347. csc(t) =

2√

3

3

64 Trigonometry


1. tan(π

4

)= 1 2. sec

(π6

)=

2√

3

33. csc

(5π

6

)= 2

4. cot

(4π

3

)=

√3

35. tan

(−11π

6

)=

√3

36. sec

(−3π

2

)is undefined

7. csc(−π

3

)= −2

√3

38. cot

(13π

2

)= 0 9. tan (117π) = 0

10. sec

(−5π

3

)= 2 11. csc (3π) is undefined 12. cot (−5π) is undefined

13. tan

(31π

2

)is undefined 14. sec

(π4

)=√

2 15. csc

(−7π

4

)=√

2

16. cot

(7π

6

)=√

3 17. tan

(2π

3

)= −√

3 18. sec (−7π) = −1

19. csc(π

2

)= 1 20. cot

(3π

4

)= −1

21. Quadrant II. 22. Quadrant III. 23. Quadrant I. 24. Quadrant IV.

25. tan(θ) =√

3 when θ =π

3+ πk for any integer k

26. sec(θ) = 2 when θ =π

3+ 2πk or θ =

5π

3+ 2πk for any integer k

27. csc(θ) = −1 when θ =3π


28. cot(θ) =

√3

3when θ =

π


29. tan(θ) = 0 when θ = πk for any integer k

30. sec(θ) = 1 when θ = 2πk for any integer k

31. csc(θ) = 2 when θ =π

6+ 2πk or θ =

5π


32. cot(θ) = 0 when θ =π


33. tan(θ) = −1 when θ =3π


34. sec(θ) = 0 has no solution.

3.4 Tangent, Cosecant, Secant, and Cotangent 65

35. csc(θ) = −1

2has no solution.

36. sec(θ) = −1 when θ = π + 2πk = (2k + 1)π for any integer k

37. tan(θ) = −√

3 when θ =2π


38. csc(θ) = −2 when θ =7π

6+ 2πk or θ =

11π


39. cot(θ) = −1 when θ =3π


40. cot(t) = 1 when t =π


41. tan(t) =

√3

3when t =

π


42. sec(t) = −2√

3

3when t =

5π

6+ 2πk or t =

7π


43. csc(t) = 0 has no solution.

44. cot(t) = −√

3 when t =5π


45. tan(t) = −√

3

3when t =

5π


46. sec(t) =2√

3

3when t =

π

6+ 2πk or t =

11π


47. csc(t) =2√

3

3when t =

π

3+ 2πk or t =

2π


66 Trigonometry

3.5 Right Triangles

3.5.1 Exercises

In Exercises 1 - 8, compute the exact value in terms of trigonometric functions and then usetechnology to estimate the value to the nearest tenth.

1. A tree standing vertically on level ground casts a 120 foot long shadow. The angle of elevationfrom the end of the shadow to the top of the tree is 21.4◦. Compute the height of the tree.

2. The broadcast tower for radio station has two enormous flashing red lights on it: one at thevery top and one below the top. From a point 5000 feet away from the base of the tower onlevel ground the angle of elevation to the top light is 7.970◦ and to the second light is 7.125◦.Compute the distance between the lights.

3. From a firetower 200 feet above level ground, a ranger spots a fire off in the distance. Theangle of depression from the horizontal to the fire is 2.5◦. How far away from the base of thetower is the fire?

4. A guy wire 1000 feet long is attached to the top of a tower. When pulled taut it makes a 43◦

angle with the ground. How tall is the tower? How far away from the base of the tower doesthe wire hit the ground?

5. When I stand 30 feet away from a tree, the angle of elevation from my eyes to the top of thetree is 50◦. The angle of depression from my eyes to the base of the tree is 10◦. What is theheight of the tree?

6. From the observation deck of a lighthouse 50 feet above the surface, a lifeguard spots a boatout on the lake sailing directly toward the lighthouse. The first sighting had an angle ofdepression of 8.2◦ and the second sighting had an angle of depression of 25.9◦. How far hadthe boat traveled between the sightings?

7. I placed a remote-controlled toy rocket 50 feet away from me on the ground and launched it.When it reached its peak, I measured the angle of elevation to be 60.9◦. What is the heightof the rocket at its peak? What is the distance between me and the rocket when it is at itspeak?

8. When I place a ladder against a shed that is 12 feet tall, the foot of the ladder makes an angleof 13.9◦ with the ground. How long is the ladder?

In Exercises 9 - 10, use a right triangle to prove the given rule. (Hint: It may be helpful to recallthat 90◦ − θ is the measure of the ‘other’ acute angle in the right triangle besides θ.)

9. cos(θ) = sin (90◦ − θ) 10. sin(θ) = cos (90◦ − θ)

3.5 Right Triangles 67


1. 120 tan (21.4◦) ft. ≈ 47.0 ft.

2. 5000 tan (7.970◦)− 5000 tan (7.125◦) ft. ≈ 75.0 ft.

3. 200 tan (87.5◦) ft. ≈ 4580.8 ft.

4. Height is 1000 sin (43◦) ft. ≈ 682.0 ft. Distance from the base is 1000 cos (43◦) ft. ≈ 731.4 ft.

5. 30 tan (50◦) + 30 tan (10◦) ft. ≈ 41.0 ft.

6. 50 tan (81.8◦)− 50 tan (64.1◦) ft. ≈ 244.0 ft.

7. Height is 50 tan (60.9◦) ft. ≈ 89.8 ft. Distance from me to the rocket is 50/ cos (60.9◦) ft.≈ 102.8 ft.

8. 12/ sin (13.9◦) ft. ≈ 50.0 ft.

9. The answer is given in the problem.

10. The answer is given in the problem.

68 Trigonometry

3.6 Trigonometric Graphs

3.6.1 Exercises

In Exercises 1 - 12, graph one period of the given function. State the period, amplitude, phaseshift, and vertical shift of the function.

1. f(t) = 3 sin(t) 2. g(t) = sin(3t) 3. h(t) = −2 cos(t)

4. f(t) = cos(t− π

2

)5. g(t) = − sin

(t+ π

3

)6. h(t) = sin(2t− π)

7. f(t) = −13 cos

(12 t+ π

3

)8. g(t) = cos(3t− 2π) + 4 9. h(t) = sin

(−t− π

4

)− 2

10. f(t) = 23 cos

(π2 − 4t

)+ 1 11. g(t) = −3

2 cos(2t+ π

3

)− 1

2 12. h(t) = 4 sin(−2πt+ π)

In Exercises 13 - 15, graph one period of the given function. State the period of the function.

13. y = tan(t− π

3

)14. y = 2 tan

(1

4t

)− 3 15. y =

1

3tan(−2t− π) + 1

3.6 Trigonometric Graphs 69


1. f(t) = 3 sin(t)Period: 2πAmplitude: 3Phase Shift: 0Vertical Shift: 0 t

y

π2

π 3π2

2π

−3

3

2. g(t) = sin(3t)Period: 2π

3Amplitude: 1Phase Shift: 0Vertical Shift: 0

t

y

π6

π3

π2

2π3

−1

1

3. h(t) = −2 cos(t)Period: 2πAmplitude: 2Phase Shift: 0Vertical Shift: 0

t

y

π2

π 3π2

2π

−2

2

4. f(t) = cos(t− π

2

)Period: 2πAmplitude: 1Phase Shift: π

2Vertical Shift: 0

t

y

π2

π 3π2

2π 5π2

−1

1

70 Trigonometry

5. g(t) = − sin(t+ π

3

)Period: 2πAmplitude: 1Phase Shift: −π

3Vertical Shift: 0

t

y

−π3

π6

2π3

7π6

5π3

−1

1

6. h(t) = sin(2t− π)Period: πAmplitude: 1Phase Shift: π

2Vertical Shift: 0

t

y

π2

3π4

π 5π4

3π2

−1

1

7. f(t) = −13 cos

(12 t+ π

3

)Period: 4πAmplitude: 1

3Phase Shift: −2π

3Vertical Shift: 0 t

y

−2π3

π3

4π3

7π3

10π3

−13

13

8. g(t) = cos(3t− 2π) + 4Period: 2π

3Amplitude: 1Phase Shift: 2π

3Vertical Shift: 4

t

y

2π3

5π6

π 7π6

4π3

3

4

5

3.6 Trigonometric Graphs 71

9. h(t) = sin(−t− π

4

)− 21

Period: 2πAmplitude: 1Phase Shift: −π

4Vertical Shift: −2

t

y

−9π4 −

7π4 −

5π4 −

3π4−π

4π4

3π4

5π4

7π4

−3

−2

−1

10. f(t) = 23 cos

(π2 − 4t

)+ 12

Period: π2

Amplitude: 23

Phase Shift: π8

Vertical Shift: 1

t

y

−3π8−π

4 −π8

π8

π4

3π8

π2

5π8

13

1

53

11. g(t) = −32 cos

(2t+ π

3

)− 1

2Period: πAmplitude: 3

2Phase Shift: −π

6Vertical Shift: −1

2t

y

−π6

π12

π3

7π12

5π6

−2

−12

1

12. h(t) = 4 sin(−2πt+ π)3

Period: 1Amplitude: 4Phase Shift: 1

2Vertical Shift: 0 t

y

−12 −

14

14

12

34

1 54

32

−4

4

1Two periods of the graph are shown in case students make different choices.2Again, we graph two periods in case students make different choices.3This will be the last time we graph two periods in case students make different choices.

72 Trigonometry

13. y = tan(t− π

3

)Period: π

t

y

−π6

π12

π3

7π12

5π6−1

1

14. y = 2 tan

(1

4t

)− 3

Period: 4π

t

y

−2π −π π 2π

−5

−3

−1

15. y =1

3tan(−2t− π) + 1

Period:π

2

t

y

− 3π4 −

5π8−π2 − 3π

8−π4

43

123

3.7 The Five Fundamental Identities 73

3.7 The Five Fundamental Identities

3.7.1 Exercises

In Exercises 1 - 35, prove the identity. Assume that all quantities are defined.

1. cos(θ) sec(θ) = 1 2. tan(t) cos(t) = sin(t)

3. sin(θ) csc(θ) = 1 4. tan(t) cot(t) = 1

5. csc(x) cos(x) = cot(x) 6.sin(t)

cos2(t)= sec(t) tan(t)

7.cos(θ)

sin2(θ)= csc(θ) cot(θ) 8.

1 + sin(x)

cos(x)= sec(x) + tan(x)

9.1− cos(θ)

sin(θ)= csc(θ)− cot(θ) 10.

cos(t)

1− sin2(t)= sec(t)

11.sin(x)

1− cos2(x)= csc(x) 12.

sec(t)

1 + tan2(t)= cos(t)

13.csc(θ)

1 + cot2(θ)= sin(θ) 14.

tan(x)

sec2(x)− 1= cot(x)

15.cot(t)

csc2(t)− 1= tan(t) 16. 4 cos2(θ) + 4 sin2(θ) = 4

17. 9− cos2(t)− sin2(t) = 8 18. tan3(t) = tan(t) sec2(t)− tan(t)

19. sin5(x) =(1− cos2(x)

)2sin(x) 20. sec10(t) =

(1 + tan2(t)

)4sec2(t)

21. cos2(x) tan3(x) = tan(x)− sin(x) cos(x) 22. sec4(t)− sec2(t) = tan2(t) + tan4(t)

23.cos(θ) + 1

cos(θ)− 1=

1 + sec(θ)

1− sec(θ)24.

sin(t) + 1

sin(t)− 1=

1 + csc(t)

1− csc(t)

25.1− cot(x)

1 + cot(x)=

tan(x)− 1

tan(x) + 126.

1− tan(t)

1 + tan(t)=

cos(t)− sin(t)

cos(t) + sin(t)

27. tan(θ) + cot(θ) = sec(θ) csc(θ) 28. csc(t)− sin(t) = cot(t) cos(t)

29. cos(x)− sec(x) = − tan(x) sin(x) 30. cos(x)(tan(x) + cot(x)) = csc(x)

31. sin(t)(tan(t) + cot(t)) = sec(t) 32.1

1− cos(θ)+

1

1 + cos(θ)= 2 csc2(θ)

74 Trigonometry

33.1

sec(t) + 1+

1

sec(t)− 1= 2 csc(t) cot(t) 34.

1

csc(x) + 1+

1

csc(x)− 1= 2 sec(x) tan(x)

35.1

csc(t)− cot(t)− 1

csc(t) + cot(t)= 2 cot(t)


36. sin(3π − 2θ) = − sin(2θ − 3π) 37. cos(−π

4 − 5t)

= cos(5t+ π

4

)38. tan(−x2 + 1) = − tan(x2 − 1) 39. csc(−θ − 5) = − csc(θ + 5)

40. sec(−6x) = sec(6x) 41. cot(9− 7θ) = − cot(7θ − 9)


42. sin(θ + π

2

)= cos(θ) 43. cos

(θ − π

2

)= sin(θ)

44. cos(θ − π) = − cos(θ) 45. sin(π − θ) = sin(θ)

46. tan(θ + π

2

)= − cot(θ) 47. sin(α+ β) + sin(α− β) = 2 sin(α) cos(β)

48. sin(α+ β)− sin(α− β) = 2 cos(α) sin(β) 49. cos(α+ β) + cos(α− β) = 2 cos(α) cos(β)

50. cos(α+β)− cos(α−β) = −2 sin(α) sin(β)

51.sin(t+ h)− sin(t)

h= cos(t)

(sin(h)

h

)+ sin(t)

(cos(h)− 1

h

)

52.cos(t+ h)− cos(t)

h= cos(t)

(cos(h)− 1

h

)− sin(t)

(sin(h)

h

)

3.7 The Five Fundamental Identities 75


The answers to the exercises are given in the problems.

76 Trigonometry

Chapter 4

Advanced Trigonometry andAlgebra

78 Advanced Trigonometry and Algebra

4.1 Arcsine, Arccosine, and Arctangent

4.1.1 Exercises

In Exercises 1 - 27, determine the exact value.

1. arcsin (−1) 2. arcsin

(−√

3

2

)3. arcsin

(−√

2

2

)4. arcsin

(−1

2

)

5. arcsin (0) 6. arcsin

(1

2

)7. arcsin

(√2

2

)8. arcsin

(√3

2

)

9. arcsin (1) 10. arccos (−1) 11. arccos

(−√

3

2

)12. arccos

(−√

2

2

)

13. arccos

(−1

2

)14. arccos (0) 15. arccos

(1

2

)16. arccos

(√2

2

)

17. arccos

(√3

2

)18. arccos (1) 19. arctan

(−√

3)

20. arctan (−1)

21. arctan

(−√

3

3

)22. arctan (0) 23. arctan

(√3

3

)24. arctan (1)

25. arctan(√

3)

26. arcsec (1) 27. arccsc (1)


28. sin

(arcsin

(1

2

))29. sin

(arcsin

(−√

2

2

))30. sin

(arcsin

(3

5

))

31. sin (arcsin (−0.42)) 32. sin

(arcsin

(5

4

))33. cos

(arccos

(√2

2

))

34. cos

(arccos

(−1

2

))35. cos

(arccos

(5

13

))36. cos (arccos (−0.998))

37. cos (arccos (π)) 38. tan (arctan (−1)) 39. tan(arctan

(√3))

40. tan

(arctan

(5

12

))41. tan (arctan (3π))

4.1 Arcsine, Arccosine, and Arctangent 79


42. arcsin(

sin(π

6

))43. arcsin

(sin(−π

3

))44. arcsin

(sin

(3π

4

))45. arcsin

(sin

(11π

6

))46. arcsin

(sin

(4π

3

))47. arccos

(cos(π

4

))48. arccos

(cos

(2π

3

))49. arccos

(cos

(3π

2

))50. arccos

(cos(−π

6

))51. arccos

(cos

(5π

4

))52. arctan

(tan

(π3

))53. arctan

(tan

(−π

4

))54. arctan (tan (π)) 55. arctan

(tan

(π2

))56. arctan

(tan

(2π

3

))



1. arcsin (−1) = −π2

2. arcsin

(−√

3

2

)= −π

33. arcsin

(−√

2

2

)= −π

4

4. arcsin

(−1

2

)= −π

65. arcsin (0) = 0 6. arcsin

(1

2

)=π

6

7. arcsin

(√2

2

)=π

48. arcsin

(√3

2

)=π

39. arcsin (1) =

π

2

10. arccos (−1) = π 11. arccos

(−√

3

2

)=

5π

612. arccos

(−√

2

2

)=

3π

4

13. arccos

(−1

2

)=

2π

314. arccos (0) =

π

215. arccos

(1

2

)=π

3

16. arccos

(√2

2

)=π

417. arccos

(√3

2

)=π

618. arccos (1) = 0

19. arctan(−√

3)

= −π3

20. arctan (−1) = −π4

21. arctan

(−√

3

3

)= −π

6

22. arctan (0) = 0 23. arctan

(√3

3

)=π

624. arctan (1) =

π

4

25. arctan(√

3)

=π

3 26. arcsec (1) = 0 27. arccsc (1) =π

2

28. sin

(arcsin

(1

2

))=

1

229. sin

(arcsin

(−√

2

2

))= −√

2

2

30. sin

(arcsin

(3

5

))=

3

531. sin (arcsin (−0.42)) = −0.42

32. sin

(arcsin

(5

4

))is undefined. 33. cos

(arccos

(√2

2

))=

√2

2

34. cos

(arccos

(−1

2

))= −1

235. cos

(arccos

(5

13

))=

5

13

4.1 Arcsine, Arccosine, and Arctangent 81

36. cos (arccos (−0.998)) = −0.998 37. cos (arccos (π)) is undefined.

38. tan (arctan (−1)) = −1 39. tan(arctan

(√3))

=√

3

40. tan

(arctan

(5

12

))=

5

1241. tan (arctan (3π)) = 3π

42. arcsin(

sin(π

6

))=π

643. arcsin

(sin(−π

3

))= −π

3

44. arcsin

(sin

(3π

4

))=π

445. arcsin

(sin

(11π

6

))= −π

6

46. arcsin

(sin

(4π

3

))= −π

347. arccos

(cos(π

4

))=π

4

48. arccos

(cos

(2π

3

))=

2π

349. arccos

(cos

(3π

2

))=π

2

50. arccos(

cos(−π

6

))=π

651. arccos

(cos

(5π

4

))=

3π

4

52. arctan(

tan(π

3

))=π

353. arctan

(tan

(−π

4

))= −π

4

54. arctan (tan (π)) = 0 55. arctan(

tan(π

2

))is undefined

56. arctan

(tan

(2π

3

))= −π

3


4.2 Inverse Trigonometric Expressions

4.2.1 Exercises

In Exercises 1 - 16, simplify each of the following functions as algebraic functions of x (withouttrigonometric expressions).

1. f(x) = sin (arccos (x)) 2. f(x) = cos (arctan (x)) 3. f(x) = tan (arcsin (x))

4. f(x) = sec (arctan (x)) 5. f(x) = csc (arccos (x)) 6. f(x) = sin (2 arctan (x))

7. f(x) = sin (2 arccos (x)) 8. f(x) = cos (2 arctan (x)) 9. f(x) = sin(arccos(2x))

10. f(x) = sin(

arccos(x

5

))11. f(x) = cos

(arcsin

(x2

))12. f(x) = cos (arctan (3x))

13. f(x) = sin(2 arcsin(7x)) 14. f(x) = sin

(2 arcsin

(x√

3

3

))

15. f(x) = cos(2 arcsin(4x)) 16. f(x) = sec(arctan(2x)) tan(arctan(2x))

4.2 Inverse Trigonometric Expressions 83


1. f(x) = sin (arccos (x)) =√

1− x2

2. f(x) = cos (arctan (x)) =1√

1 + x2

3. f(x) = tan (arcsin (x)) =x√

1− x2

4. f(x) = sec (arctan (x)) =√

1 + x2

5. f(x) = csc (arccos (x)) =1√

1− x2

6. f(x) = sin (2 arctan (x)) =2x

x2 + 1

7. f(x) = sin (2 arccos (x)) = 2x√

1− x2

8. f(x) = cos (2 arctan (x)) =1− x2

1 + x2

9. f(x) = sin(arccos(2x)) =√

1− 4x2

10. f(x) = sin(

arccos(x

5

))=

√25− x2

5

11. f(x) = cos(

arcsin(x

2

))=

√4− x2

2

12. f(x) = cos (arctan (3x)) =1√

1 + 9x2

13. f(x) = sin(2 arcsin(7x)) = 14x√

1− 49x2

14. f(x) = sin

(2 arcsin

(x√

3

3

))=

2x√

3− x23

15. f(x) = cos(2 arcsin(4x)) = 1− 32x2

16. f(x) = sec(arctan(2x)) tan(arctan(2x)) = 2x√

1 + 4x2


4.3 Trigonometric Equations

4.3.1 Exercises

In Exercises 1 - 18, compute all of the exact solutions of the equation and then list those solutionswhich are in the interval [0, 2π).

1. sin (5θ) = 0 2. cos (3t) =1

23. sin (−2x) =

√3

2

4. tan (6θ) = 1 5. csc (4t) = −1 6. sec (3x) =√

2

7. cot (2θ) = −√

3

38. cos (9t) = 9 9. sin

(x3

)=

√2

2

10. cos

(θ +

5π

6

)= 0 11. sin

(2t− π

3

)= −1

212. 2 cos

(x+

7π

4

)=√

3

13. csc(θ) = 0 14. tan (2t− π) = 1 15. tan2 (x) = 3

16. sec2 (θ) =4

317. cos2 (t) =

1

218. sin2 (x) =

3

4

In Exercises 19 - 28, compute the exact solutions of the equation which are in the interval [0, 2π).

19. sin (θ) = cos (θ) 20. sin (2t) = sin (t)

21. sin (2x) = cos (x) 22. cos (2θ) = sin (θ)

23. cos (2t) = cos (t) 24. cos(2x) = 2− 5 cos(x)

25. tan2(t) = 1− sec(t) 26. cot2(x) = 3 csc(x)− 3

27. tan2 (θ) =3

2sec (θ) 28. cos3 (t) = − cos (t)

4.3 Trigonometric Equations 85


1. θ =πk

5; θ = 0,

π

5,2π

5,3π

5,4π

5, π,

6π

5,7π

5,8π

5,9π

5

2. t =π

9+

2πk

3or t =

5π

9+

2πk

3; t =

π

9,5π

9,7π

9,11π

9,13π

9,17π

9

3. x =2π

3+ πk or x =

5π

6+ πk; x =

2π

3,5π

6,5π

3,11π

6

4. θ =π

24+πk

6; θ =

π

24,5π

24,3π

8,13π

24,17π

24,7π

8,25π

24,29π

24,11π

8,37π

24,41π

24,15π

8

5. t =3π

8+πk

2; t =

3π

8,7π

8,11π

8,15π

8

6. x =π

12+

2πk

3or x =

7π

12+

2πk

3; x =

π

12,7π

12,3π

4,5π

4,17π

12,23π

12

7. θ =π

3+πk

2; θ =

π

3,5π

6,4π

3,11π

6

8. No solution

9. x =3π

4+ 6πk or x =

9π

4+ 6πk; x =

3π

4

10. θ = −π3

+ πk; θ =2π

3,5π

3

11. t =3π

4+ πk or t =

13π

12+ πk; t =

π

12,3π

4,13π

12,7π

4

12. x = −19π

12+ 2πk or x =

π

12+ 2πk; x =

π

12,5π

12

13. No solution

14. t =5π

8+πk

2; t =

π

8,5π

8,9π

8,13π

8

15. x =π

3+ πk or x =

2π

3+ πk; x =

π

3,2π

3,4π

3,5π

3

16. θ =π

6+ πk or θ =

5π

6+ πk; θ =

π

6,5π

6,7π

6,11π

6

17. t =π

4+πk

2; t =

π

4,3π

4,5π

4,7π

4

18. x =π

3+ πk or x =

2π

3+ πk; x =

π

3,2π

3,4π

3,5π

3


19. θ =π

4,5π

420. t = 0,

π

3, π,

5π

3

21. x =π

6,π

2,5π

6,3π

222. θ =

π

6,5π

6,3π

2

23. t = 0,2π

3,4π

324. x =

π

3,5π

3

25. t = 0,2π

3,4π

326. x =

π

6,5π

6,π

2

27. θ =π

3,5π

328. t =

π

2,3π

2

4.4 Limits 87

4.4 Limits

4.4.1 Exercises

In Exercises 1 - 12, use the graph of f(x) below to compute the given value.

-4 -3 -2 -1 1 2 3 4

-3

-2

-1

1

2

3

Graph of f(x)

1. f(0) 2. limx→0

f(x) 3. f(−1)

4. limx→−1

f(x) 5. f(1) 6. limx→1

f(x)

7. f(−3) 8. limx→−3

f(x) 9. f(3)

10. limx→3

f(x) 11. f(−4) 12. limx→−4

f(x)

In Exercises 13 - 24, use the graph of f(x) above to compute the given value.

13. limx→0−

f(x) 14. limx→0+

f(x) 15. limx→−1−

f(x)

16. limx→−1+

f(x) 17. limx→1−

f(x) 18. limx→1+

f(x)

19. limx→−3−

f(x) 20. limx→−3+

f(x) 21. limx→3−

f(x)

22. limx→3+

f(x) 23. limx→∞

f(x) 24. limx→−∞

f(x)


In Exercises 25 - 36, use the graph of g(x) below to compute the given value.

-3 -2 -1 1 2 3 4 5

-2

-1

1

2

3

4

Graph of g(x)

25. g(2) 26. limx→2

g(x) 27. g(3)

28. limx→3

g(x) 29. g(0) 30. limx→0

g(x)

31. g(−2) 32. limx→−2

g(x) 33. g(4)

34. limx→4

g(x) 35. g(−3) 36. limx→−3

g(x)

In Exercises 37 - 48, use the graph of g(x) above to compute the given value.

37. limx→2−

g(x) 38. limx→2+

g(x) 39. limx→3−

g(x)

40. limx→3+

g(x) 41. limx→0−

g(x) 42. limx→0+

g(x)

43. limx→−2−

g(x) 44. limx→−2+

g(x) 45. limx→4−

g(x)

46. limx→4+

g(x) 47. limx→∞

g(x) 48. limx→−∞

g(x)

4.4 Limits 89

In Exercises 49 - 66, use a graph to compute the given value.

49. limx→0+

1

x50. lim

x→0−

1

x51. lim

x→0

1

x

52. limx→∞

1

x53. lim

x→−∞

1

x54. lim

x→0+

1

x2

55. limx→0−

1

x256. lim

x→0

1

x257. lim

x→∞

1

x2

58. limx→−∞

1

x259. lim

x→∞ex 60. lim

x→−∞ex

61. limx→1

ln (x) 62. limx→∞

ln (x) 63. limx→0+

ln (x)

64. limx→0+

√x 65. lim

x→0−

√x 66. lim

x→∞

√x



1. f(0) = −2 2. limx→0

f(x) = 2 3. f(−1) = 1

4. limx→−1

f(x) = 0 5. f(1) = −1 6. limx→1

f(x) DNE

7. f(−3) is undefined 8. limx→−3

f(x) =∞ 9. f(3) is undefined

10. limx→3

f(x) DNE 11. f(−4) = 2 12. limx→−4

f(x) = 2

13. limx→0−

f(x) = 2 14. limx→0+

f(x) = 2 15. limx→−1−

f(x) = 0

16. limx→−1+

f(x) = 0 17. limx→1−

f(x) = 3 18. limx→1+

f(x) = −1

19. limx→−3−

f(x) =∞ 20. limx→−3+

f(x) =∞ 21. limx→3−

f(x) = −∞

22. limx→3+

f(x) =∞ 23. limx→∞

f(x) = 1 24. limx→−∞

f(x) =∞

25. g(2) = 0 26. limx→2

g(x) = 4 27. g(3) = 3

28. limx→3

g(x) = 3 29. g(0) = −1 30. limx→0

g(x) DNE

31. g(−2) is undefined 32. limx→−2

g(x) DNE 33. g(4) = −2

34. limx→4

g(x) =∞ 35. g(−3) = 3 36. limx→−3

g(x) = 3

37. limx→2−

g(x) = 4 38. limx→2+

g(x) = 4 39. limx→3−

g(x) = 3

40. limx→3+

g(x) = 3 41. limx→0−

g(x) = −1 42. limx→0+

g(x) = 1

43. limx→−2−

g(x) =∞ 44. limx→−2+

g(x) = −∞ 45. limx→4−

g(x) =∞

46. limx→4+

g(x) =∞ 47. limx→∞

g(x) = 2 48. limx→−∞

g(x) = 3

49. limx→0+

1

x=∞ 50. lim

x→0−

1

x= −∞ 51. lim

x→0

1

xDNE

52. limx→∞

1

x= 0 53. lim

x→−∞

1

x= 0 54. lim

x→0+

1

x2=∞

4.4 Limits 91

55. limx→0−

1

x2=∞ 56. lim

x→0

1

x2=∞ 57. lim

x→∞

1

x2= 0

58. limx→−∞

1

x2= 0 59. lim

x→∞ex =∞ 60. lim

x→−∞ex = 0

61. limx→1

ln (x) = 0 62. limx→∞

ln (x) =∞ 63. limx→0+

ln (x) = −∞

64. limx→0+

√x = 0 65. lim

x→0−

√x DNE 66. lim

x→∞

√x =∞

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Exercises in Math 110: Algebra and Trigonometry for Calculus