Common Core Math 3 Unit 5 Day 1 “Inverses”

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Common Core Math 3

Notes – Unit 5 Day 1

“Inverses”

Part 1:

1. Graph the function f(x) = x2 by using a table of

values.

X Y

2. Switch the ordered pairs found it step 1.

X Y

3. Graph the line y = x.

Fold your paper on the line y = x and what can you conclude?

2

Graph the inverse of each function below.

1. 2.

3

Part 2:

Steps for determine the inverse equation for a function or relation algebraically.

1. Switch the x and y. Remember that f(x) is notation for the output just like Y.

2. Solve for y to undo the operations.

3. Use -1 to denote the inverse.

Find the inverse of each function or relation.

1. (1,3), (1, 1), (1, 3), (1,1)f

2. 2 1

( )3

xg x

3.

1f( ) 2

2x x

4. 2( ) ( 2) 3g x x 5. 2( ) 9h x x

Part 3:

Composition is a new operation that is read from right to left. Notations are:

( ( ))

( ( ))

( )

( )

f g x

g f x

f g x

g f x

Examples:

1.

Find ( ( )).

( ) 1

( ) 3

f g x

f x x

g x x

2.

Find g( ( )).

( ) 8

( ) 2 3

f x

f x x

g x x

4

3. 2

Find ( ( )).

( ) 3 1

( ) 2

f g x

f x x

g x x

4. 2

Find ( ( )).

( ) 6 2

( ) 1

g f x

f x x

g x x

5. 2

Find ( (8)).

( ) 2 4

( ) 7

g f

f x x

g x x

6. 3

Find ( ( 2)).

( ) 2

( ) 2

f g

f x x

g x x

Part 4:

VERIFYING INVERSE FUNCTIONS

You can use composition of functions to determine if two functions are inverses of each other.

If ( ( )) and ( ( ))f g x g f x BOTH equal X then they are inverses of each other.

Example:

1. Determine whether the two functions are inverses of each other.

1

( ) 3 6 its inverse is g( ) 23

f x x x x

( ( ))f g x ( ( ))g f x

5

Use composition to determine whether the functions are inverses of each other.

2.

( ) 2x 3

( ) 2 3

f x

g x x

3.

( ) 6 3

3( )

6

f x x

xg x

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Common Core Math 3

Notes - Unit 5 Day 2

“Exponential Functions”

Part 1:

Exponential Functions & Their Applications

An exponential function is any positive not equal to one raised to some valued exponent. ( ) xf x a b

The shape of the curve will always be:

Classify as growth or decay and circle what determines this classification.

A. 1

25

x

y

B. 0.5(4)xy C. 7(1.2)xy D. 7 xy

Consider the parent equation ( ) xf x a b , and transformed equation ( ) x hf x a b k .

IN GENERAL, the following transform the graph of by:

h shifts: ________________ k shifts: ________________ ( )f x reflects: ________________

Identify the parent function and the transformation of the graph of each equation.

1. 2( ) 2 3 1xf x 2.

41

( ) 3 32

x

f x

3. ( ) 5 7xf x

Write the exponential equation given the following transformations.

4. ( ) 4 shift up 5 and left 4

15. ( ) shift down 2, right 3 and reflect

3

x

x

f x

f x

Exponential Growth

Where b > 1

Increasing Function

Exponential Decay

Where 0< b < 1 or negative exponent

Decreasing Function

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Ex 1: Graph the exponential function: 1( ) 4 3xf x

Equation of Parent Function A B Growth or Decay Transformations Y intercept Equation of Asymptote Domain Range Left Behavior Right Behavior

Ex 2: Graph the exponential function: 2

1( ) 5

2

x

f x


Ex 3: Graph the exponential function: 2( ) 2 1xf x


x y

x y

x y

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Part 2: The Effects of A and B

To get a better feeling for the effect of a and b on the graph, examine the sets of graphs below. The

first set shows various graphs, where a remains the same and we only change the value for b.

Example A:

A B C D E F

Notice that changing the value for a changes the vertical (y) intercept. Since a is multiplying the bx

term, a acts as a vertical stretch factor, not as a shift. Notice also that the end behavior for all of

these functions is the same because the growth factor did not change and none of these a values

introduced a vertical flip.

Example B:

Match the graph to the equation. A B C D

1. ( ) 2(1.3)

2. ( ) 2(1.8)

3. ( ) 4(1.3)

4. ( ) 4(0.7)

x

x

x

x

f x

g x

h x

k x

Write the letter from the graph on the left that

matches the equation.

1. .8

2. 1.6

3. 1.8

4. .4

5. 2.3

6. 3.1

x

x

x

x

x

x

y

y

y

y

y

y

Summarize:

Growth: Decay:

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Practice:

A.

B.

C.

Match the letter from the

graph on the right to the

equation.

1. ( ) (0.8)

2. ( ) 2(1.9)

3. ( ) 3(0.7)

4. ( ) 3(0.3)

5. ( ) 5(0.9)

6. ( ) 2(1.5)

x

x

x

x

x

x

f x

g x

h x

k x

a x

b x

Write the equation next to

the matching graph.

1. ( ) 0.6(0.7)

2. ( ) 2(3.1)

3. ( ) (1.7)

4. ( ) 0.8(0.9)

x

x

x

x

f x

g x

h x

k x

Match the equation next to the

matching graph.

𝐴. 𝒚 = 𝟖𝒙

𝑩. 𝒚 = 𝟏. 𝟐𝒙

𝑪. 𝒚 = 𝟏𝒙

𝑫. 𝒚 = 𝟐𝒙

𝑬. 𝒚 = 𝟎. 𝟐𝒙

1 2 3

4

5

A B

C D

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Part 3: Simplifying Exponential Expressions

Laws of Exponents

1) a a am n m n 2) m

m n

n

aa

a

3) a 0 1

4) a amn

m n 5) ab a bn n n

6) a

b

a

b

n n

n

n na b

b a

n m

m n

a b

b a

7) 1n

na

a

1 n

na

a

1. a5a1 = 3. (-2x3)4 2. (b3b5)0 4. (-3x2y)3

5. (xy)-1 7. (𝑥

𝑦)

2

6. (3𝑥

𝑦)

−2

8. (𝒙𝟐

𝒂𝒙−𝟑)−𝟑

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Common Core Math 3


Log Functions

A. Log Functions - The inverse of an exponential

Two Forms:

1. Log Form ________________________2. Exponential Form________________________.

Write the equation in exponential form.

1. 2log 8 3 2. 5log 25 2 3. 4log 1 0

Write the exponential equation in log form.

1. 42 16 2. 1

29 3 3. 310 1000

B. Evaluate a Log Expression

Base 10 Log

1. 10log 99 2. 10log 5 3. log 20

Not a base of 10

Formula to Use:

1. 2log 16 2. 3log 7 3. 11log 20

C. NO Calculator.

1. 4log 64 2. 5log 25 3. 3

1log

3

4. 9log 1 5. 2log 16 6. 2

1log

16

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D. Inverse Property of Exponents

1. 8

6log 6 2. 2

5log 5 3. 3log (4 1)3

x 4. 4log (2 3)

4x

E. Solving Log Equations

Use the above information to solve.

1. 2log 8 9x 2.

2

3log 3 16x 3. 7log (3 1)

7 14x

F. Graph the following log functions.

Find the inverse of the log equation (which will be an exponential equation).

Graph the exponential equation.

Graph the inverse of the exponential equation.

1. 5( ) logf x x 2. 3( ) log ( 1)f x x 3. ( ) log 2f x x

Inverse: Inverse: Inverse:

Exponential Graph:

Exponential Graph:

Exponential Graph:

Log Graph:

Log Graph:

Log Graph:

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Common Core Math 3


“Properties of Logs and Solving Log Equations”

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15

Condense each log into a single log.

1. 2log 4 3log 2x 2. 1

log16 log 22

3. 5log( 1) 3logx x 4. log5 (2log3 3log( 2))x x x

5. log5 log 2 log( 3)x x x 6. 2 2 2log 4 log 2 log 8

Part IV

Solving a Log Equation

A, Logs on Both Sides

1. 2 2log (3 5) log ( 7)x x 2. 2

5 5log (3 1) log 2x x

3. 2

3 3log ( 2) logp p 4. 7 7 7log ( 2) log ( 4) log 15x x

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5. 3 3 3log ( 3) log (4 1) log 5c c 6. 2

5 5 5log 6 log 2 log 48x

7. 9 9 9log ( 6) log log 2x x

B. Log on One Side

1. 4

5log

2n 2. 9

3log

2x 3. 3log (2 1) 2x

7. 5 5log ( 2) log ( 4) 2c c 9. 2 2log ( 4) log ( 3) 3x x

9. 4 4log ( 1) 1 log ( 2)n n 10. 2 2log (9 5) 2 log ( 1)x x

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Common Base Math 3

Notes – Unit 5 Day 5

“Exponential Equations”

Part 1: Common Base

Solve each exponential equation.

1. 2 13 81n 2.

5 2 13 9x x

3. 2 14 8x x 4.

9 24 256n

5. 3 1 1

4256

p 6. 3 2 1

5625

k

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Part 2: Not a Common Base

Write your answer as a log expression and then evaluate.

Examples:

1. 3 11x 2. 22 82.1a

3. 29 38m 4.

2 14 9x x

5. 5 7 36 2x x 6.

5 22 3x x

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Common Core Math 3


“Natural Logs”

I. Natural Log

Symbol_________________ Definition_____________________________

Example:_______________________ is the same as _____________________

“e” is a symbol (like pi) that represents _______________

Rule:____________________________

Evaluate:

A. 2e B. 2e C. ln 4

D. ln5 1 E. 4 3ln xe F. 2ln6

G. ln 21e H. 2 1ln xe I. 8lne

II. Solving Equations. Write your answer as a log expression and then evaluate.

A. 5xe B. 5 7 2xe

C. 23 4 10xe D. ln3 0.5x

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E. ln(2 3) 2.5x F. 23 8xe

G. 2ln3 1 5x H. ln5 ln 7x x I. ln7 ln5 ln70x

Review of all Equations..

1) 2 2log ( 2) log 5 4x 2) 4 4log (2 1) log ( 2) 1x x

3) 4 4 4log ( 4) log log 5x x 4) ln( 1) 2x

5) 2

2 2log ( 8) log log 6x x x 6) 3 3log log ( 2) 1x x

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7) ln( 2) 3x 8) 2ln 8x

9) log 64 3x 10) 3ln 18x

11) ln 2x

e 12)

1ln 5

x

13) ln ln( 1) ln 2x x 14) 2(ln ) 16x

15) ln ln( 3) ln10x x 16) 6log ( 2) 2x

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Common Core Math 3


“Application of Exponentials and Logs”

Formulas:

1

ntr

A Pn

Use when ________________________________________

(1 )tA P r Used when____________________________________

rtA Pe Used when_____________________________________

1

2

t

k

A P

Used when_____________________________________

1 (1 ) n

iPA

i

Used when___________________________________________

A = Ending Amount

P = Initial/Beginning Amount

r = Rate

t = Time

n = Number of times principal is compounded

k = Half life of the sample

i is the monthly interest rate (r/12)

1. $800 is deposited into an account that pays 9% annual interest, compounded quarterly. Find the

balance after four years.

2. You have inherited an emerald ring that had an appraised value of $2400 in 1960. The appraised

value of the ring has increased by approximately 6% each year. What is the value in 2011?

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3. $600 is deposited in an account that pays 6.5% interest, compounded continuously. What is the

balance after 5 years?

4. You just bought a $230,000 house, with 10% down on a 30-year mortgage with an interest rate of

8.5% per year. What is the monthly payment?

5. How long will it take an investment to triple in an account that pays 8.5% annual interest compounded

continuously?

6. For Dave to buy a new car comparably equipped to the one he bought years ago would cost $12,500.

Since Dave bought the car, the inflation rate for cars like his has been at an average annual rate of

5.1%. If Dave originally paid $8400 for the car, how long ago did he buy it?

7. How much must you deposit into an account that pays 6.25% interest, compounded semiannually, to

have a balance of $1000 after 2 years?

8. Dawn is saving money to go to Europe after her college graduation. She will finish college 6 years

from now. If the six year certificate of deposit that she buys now is compounded continuously and she

puts in $1500, at what rate should she look for to have a total of $3000 for her trip?

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9. A particular isotope has a half-life of 6 minutes. How much of this isotope is left after 58 minutes if

you had 800 grams to start with?

10. The half-life of radium-226 is 1200 years. Suppose we have a 52 mg sample. Find a function that

models the mass remaining after t years. How much of the sample will remain after 500 years. After

how long will only 18 mg of the sample remain?

11. Find the monthly payment for the following home in Hawaii.

12. Find the monthly payment for a home that is listed for $250,500, at a rate of 3.38% for 30 years

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Common Core Math 3 Unit 5 Day 1 “Inverses”