1

Math 165 – Section 2.1 – Functions Name _________________________ Read the book or the power point presentations for this section. Complete for next class: ONLY PAGE 1 AND #10, 11, AND 12 ON PAGE 4 - the rest will be done in class

1) What is a function?

2) Give an example of a “table” which a) Is NOT a function b. IS a function

3) (This is in section 2.2 ) - Give an example of a “graph” which is a) Is NOT a function b. IS a function

4) Sketch the graph of each of the following and indicate whether they represent a function or not. a) Y = 2x + 3 b) 2 1y x= + c) 2 2 6x y+ =

Section 2.1 – Evaluating Functions

5) Given the functions ( ) 3 2f x x= − and 2( ) 5 2g x x x= − + ; find each of the following – show work a) g(-1)

b) g(a)

c) f(x+h)

2

Section 2.1 – continued - Evaluation 6) If a rock falls from a height of 20 meters on the planet Jupiter, its height 𝐻𝐻 (in meters)

after 𝑥𝑥 seconds is approximately 𝐻𝐻(𝑥𝑥) = 20 − 13𝑥𝑥2. a) Find 𝐻𝐻(0) and 𝐻𝐻(1.2). Interpret your answers in the context of the situation

using a complete and grammatically correct sentence.

b) Solve H(x) = 15 and interpret in context.

c) When does the rock strike the ground?

Section 2.1 - Operations with Functions

7) Given the functions ( ) 3 2f x x= − and 2( ) 5 2g x x x= − + ; find each of the following – show work a) (f+g)(x)

b) f(x)-g(x)

c) (f•g)(-1)

3

Section 2.1 – continued - Operations 8) #113, page. 71, section 2.1 - When the driver of a vehicle observes an impediment, the total stopping distance

involves both the reaction distance (the distance the vehicle travels while the driver moves his or her foot to the brake pedal) and the braking distance (the distance the vehicle travels once the brakes are applied). For a car traveling at a speed of v miles per hour, the reaction distance R, in feet, can be estimated by 𝑅𝑅(𝑣𝑣) =2.2𝑣𝑣. Suppose that the braking distance B, in feet, for a car is given by

𝐵𝐵(𝑣𝑣) = 0.05𝑣𝑣2 + 0.4𝑣𝑣 − 15 a) Find the stopping distance function D(v) = (R+B)(v) b) Find D(60) and interpret in context.

9) Given 𝑓𝑓(𝑥𝑥) = 2𝑥𝑥−13𝑥𝑥+7

and 𝑔𝑔(𝑥𝑥) = 𝑥𝑥2−2𝑥𝑥+53𝑥𝑥+7

; find (f – g)(x)

4

Section 2.1 – Evaluating Functions and Domain 10) If ( ) 7f x x= + ; evaluate the function for the following values of x; round to one decimal place if

needed. If it is not a real number, say so.

x -2 6 -7 -8 -1 -9 f (x)

11) 2( )5

g xx

=−

; evaluate the function for the following values of x; leave answers as fractions or round to

one decimal place. If it is undefined, say so.

x 7 -2 5 3 8 3.2 g(x)

12) 2( )4

xh xx−

=−

; evaluate the function for the following values of x; round to one decimal place. If it is

undefined, or it is not a real number, say so.

x 7 2 0 4 8 1.2 g(x)

13) Give the domain of the following functions. Domain: What numbers can you assign to x so that you get a real number answer for y? What are the x-coordinates of the points on the graph of the function?

a) Square root function: ( ) 7f x x= +

b) Rational function: 2( )5

g xx

=−

c) Mixed function: 2( )4

xh xx−

=−

5

Section 2.1 – continued - Domain 14) Find the domain of each function. State the domain using interval notation.

𝑓𝑓(𝑥𝑥) = 𝑥𝑥+2𝑥𝑥2−144

b) 𝑔𝑔(𝑥𝑥) = 3𝑥𝑥 − 5 c) ℎ(𝑥𝑥) = √2𝑥𝑥 + 5

15) Summary of domain – explain procedures for each type of function a) Linear and quadratic functions

b) Rational functions c) Square root functions

16) Identify the domain and range of the functions whose graphs are shown. State your answer using interval notation.

6

Section 2.1 - continued – Difference Quotient 17) Difference Quotient

18) Given the function 2( ) 7 5f x x x= − + ; find the difference quotient 𝑓𝑓(𝑥𝑥+ℎ)−𝑓𝑓(𝑥𝑥)ℎ

and give the geometric interpretation of the obtained expression.

19) Write a quadratic function and find the difference quotient.

20) Write a linear function and find the difference quotient.

7

Math 165 – Section 2.2 – Graphs of Functions Name ________________ Read the book or the power point presentations for these sections.

21) Use the graph of y = f(x) to answer the questions

a) What is the domain?

b) What is the range?

c) For what values of x is f(x) = 0?

d) Solve f(x) = -2?

e) What is x when f(x) = 4?

f) Give intervals where the function is decreasing

g) Give intervals where the function is constant

h) Give intervals where the function is increasing

i) For what values of x is f(x) negative?

8

22) Problem # 31, page 78, section 2.2 –

Given that f(x) is the linear function and g(x) is the quadratic function, use the graph to find each of the following:

a) (f + g) (2)

b) (f + g)(4)

c) (f – g) (2)

d) (g – f) (0)

e) (f●g)(2)

f) (𝑓𝑓𝑔𝑔

)(4)

9

Math 165 – Section 2.3 – Increasing and Decreasing – Local and Absolute Extrema

23) For the function f(x) = 𝑓𝑓(𝑥𝑥) = 𝑥𝑥4 − 4𝑥𝑥2 + 𝑥𝑥 + 5,𝑢𝑢se a graphing utility to determine a) Intervals of increase b) Intervals of decrease c) Local and absolute extrema

24) Find the absolute maximum and the absolute minimum, if they exist.

25) Write the Extreme Value Theorem

10

Section 2.3 – Even and Odd Functions

26) Given 𝑓𝑓(𝑥𝑥) = 3𝑥𝑥4 − 𝑥𝑥2 − 4, find f(-x). What do you notice? Explore the graph and specify the type of symmetry.

27) Make up another function with the same characteristics.

28) Given 𝑓𝑓(𝑥𝑥) = 𝑥𝑥3 − 𝑥𝑥, find f(-x). What do you notice? Explore the graph and specify the type of symmetry.

29) Make up another function with the same characteristics.

30) A function is even if for every (x) in the domain, (- x) is also in the domain and

31) A function is odd if for every (x) in the domain, (- x) is also in the domain and

32) An even function has _____________ symmetry.

33) An odd function has ______________ symmetry.

11

Section 2.3 – Average Rate of Change

34) Find the average or change and the equation of the secant line

( ) 2Suppose that 2 4 3.g x x x= − + −

12

Math 165 – Section 2.4 – Library of Functions Name ________________

Read the book or the power point presentations for these sections.

35) For each of the basic elementary functions, graph and give domain and range

Function Graph Domain Range Constant function

Identity function

Square function

Cube function

Square root function

Cube root function

Reciprocal function

Absolute value function

Greatest integer function

13

Section 2.4 – Piecewise defined Function

36) A function that is defined by two (or more) equations over a specified domain is called a piecewise function.

1. ( )f x x= =

2. 2 2

( )0 2

x if xg x

if x≠

= =

3.

2 4, 1( ) 4, 1 3

, 3

x xh x x

x x

+ ≤= < ≤− >

ℎ(−5) =

ℎ(8) =

ℎ(2) =

4. The graph of a piecewise–defined function is given. Write a formula that defines the function.

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

2 3 4 5 1

1 2 3 4

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

14

37) Section 2.4 - Making up our own piecewise defined problems – from function to graph and from graph to function. a)

b)

c)

d)

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

-1 2 3 4 5 1 -2 -3 -4 -5

1

2

3

4

-1 -2

-3

15

Section 2.5 – Transformations of Functions

38) Explore the Vertical Shift of a Function y = f(x) On the same coordinate system, show the graph of the family of functions y = f(x) + k:

a. 𝑦𝑦 = |𝑥𝑥| b. 𝑦𝑦 = |𝑥𝑥| + 2 c. 𝑦𝑦 = |𝑥𝑥| − 3

Conclusion: (will be done in class) • If k > 0, then the graph of y = f(x) + k is the graph of y = f(x) shifted _________________ ______ k units.

• If k < 0, then the graph of y = f(x) + k is the graph of y = f(x) shifted _________________ ______ k units.

39) Explore the Horizontal Shift of a Function y = f(x)

On the same coordinate system, show the graph of the family of functions y = f(x - h):

a) 𝑦𝑦 = √𝑥𝑥 b) 𝑦𝑦 = √𝑥𝑥 − 2 c) 𝑦𝑦 = √𝑥𝑥 + 3

Conclusion: (will be done in class) • If h > 0, then the graph of y = f(x - h) is the graph of y = f(x) shifted _________________ ______ h units.

• If h < 0, then the graph of y = f(x - h) is the graph of y = f(x) shifted _________________ ______ h units.

16

40) Explore the Vertical Stretch and Compression of a Function y = f(x) On the same coordinate system, show the graph of the family of functions y = a f(x):

a) 𝑦𝑦 = 𝑥𝑥2 b) 𝑦𝑦 = 2𝑥𝑥2 c) 𝑦𝑦 = 1

2𝑥𝑥2

Conclusion: (will be done in class) • If a > 1, then the graph of y = f(x) has been (stretched / compressed) vertically. (circle one choice).

• If 0 < a < 1, then the graph o f y = f(x) has been (stretched / compressed) vertically. (circle one choice).

41) Explore the x-reflection of a Function y = f(x)

On the same coordinate system, show the graph of the family of functions y = a f(x):

a) 𝑦𝑦 = 𝑥𝑥2 b) 𝑦𝑦 = −𝑥𝑥2

Conclusion: (will be done in class) • If a < 0, then the graph of y = f(x) has been reflected with respect to the _________ axis.

17

Section 2.6 - Mathematical Models; Building Functions

42) Examples:

1. A rectangle is inscribed in a circle of radius 3 centered at the origin. Let ( , )P x y= be a point in quadrant I that is a vertex of the rectangle and is on the circle. Express the area A of the rectangle as a function of x and the perimeter P of the rectangle as a function of x. Find the maximum area. For what value of 𝑥𝑥 is the area the largest? For what value of 𝑥𝑥 is the perimeter the largest?

2. Let ( , )P x y= be a point on the graph of 3y x= . Express the distance d from P to the

point (2, 0) as function of x. What is d if 1x = − ? For what values of 𝑥𝑥 is the distance 𝑑𝑑 the smallest?

3. A right triangle has one vertex on the graph of 216y x= − , 0x > , at ( , )x y , another at

the origin, and the third on the positive x-axis at ( ,0)x . Express the area A of the

triangle as a function of x.

4. An open box with a square base is to be made from a square piece of cardboard 18 inches on a side by cutting out a square from each corner and turning up the sides. Express the volume V of the box as a function of the length x of the side of the square cut from each corner. Find the dimensions of the box of maximum volume. What is the maximum volume?

5. A media company is going to install cable from a house to their connection box B. The house is located at one end of a driveway 7 miles back from a road (see diagram). The other end of the driveway and the nearest connection box are on the same road, 25 miles apart. The cost of installing the cable is $656 per mile off the road and $375 per mile along the road. Let x be the distance from where the driveway meets the road to where the cable comes to the road. Develop a function C(x) that expresses the total installation cost as a function of x.Now use your calculator to graph C. Use the graph to determine the value of x that will produce the minimum cost. Round to the nearest thousandth of a mile.

(Tip: use a window [0, 20, 1] x [ 12,000 , 20,000 , 1000 ].)

State the minimum cost for that installation, rounded to the nearest cent.

25 mi

7 mi B

cable

cable x