Unit 6 Packet: Operations, Composition, Inverses Name________________________________________Period_____

Schedule and Homework

2CP Lesson: Operations with Functions

Core Concepts

Operations on Functions

Let f and g be any two functions. A new function can be defined by performing any of the four basic operations on f

and g.

The domains of the sum, difference, product, and quotient functions consist of the x-values that are in the domains of both

f and g. Additionally, the domain of the quotient does not include x-values for which 0.g x

22-Jan Operations with Functions WS Operations of Functions

23-Jan Composition of Functions WS Composition of Functions (Odds Only)

24-Jan Inverse of a Function: equations and graphing, one-to-one, etc WS Inverse Functions #1-10

25-Jan Inverse of a Function: proving inverses WS Inverse Functions #11-15

28-Jan Review Study for Test

29-Jan Half-Test- 50 points None

Operation Definition Example: f x x g x x( ) 5 , ( ) 2

Addition f g x f x g x 5 2 6 2f g x x x x

Subtraction f g x f x g x 5 2 4 2f g x x x x

Multiplication fg x f x g x 25 2 5 10fg x x x x x

Division

f xfx

g g x

5

2

f xx

g x

Let’s Try:

In Exercises 1–2, find f g x f g x and and state the domain of each. Then evaluate

f g f g and for the given value of x.

1. 2 23 8, 6 3 ; 1f x x x g x x x x

2. 𝑓(𝑥) = 3√𝑥 + 2, 𝑔(𝑥) = −2√𝑥 − 5; 𝑥 = 16

In Exercises 3-4, find fg x and f xg

and state the domain of each. Then evaluate fg and fg

for the given

value of x.

3.𝑓(𝑥) = 𝑥2 + 5𝑥 − 2, 𝑔(𝑥) = 3𝑥 − 2; 𝑥 = −2 4. 1 212 , 11 ; 4f x x g x x x

In Exercise 5, find (𝟐𝒇 + 𝒈)(𝒙)𝒂𝒏𝒅 (−𝒇 − 𝒈)(𝟑), 𝒈𝒊𝒗𝒆𝒏 𝒇(𝒙) = 𝟐𝒙 − 𝟏 𝒂𝒏𝒅 𝒈(𝒙) = 𝟓𝒙

In Exercises 6, use the table to state the domain of f(x) and g(x). Then find the following:

a. (𝑓 + 𝑔)(1)

b. (𝑓𝑔)(1)

c. (𝑓

𝑔) (0)

d. 𝑓(5) − 𝑔(9)

e. (2𝑔 + 𝑓)(−3)

In Exercise 7, use the graph to state the domain of f(x) and g(x). Then find the following:

a. (𝑓 + 𝑔)(1) d. 𝑓(5) − 𝑔(9)

b. (𝑓𝑔)(1)

c. (𝑓

𝑔) (0) e. (2𝑔 + 𝑓)(−3)

Apply Operations with Functions Operations with functions can apply to real-world situations.

Example: The players on a basketball team participated in a fundraiser and raised $580 to help pay

for shoes for each team member. The shoes cost $100 each, and there is a shipping and handling fee of

$50 on each order. Sales tax of 6% is charged on the entire bill. The team member that raised the

most money in the fundraiser does not have to pay for her shoes. The remaining players will split the

remaining cost evenly. Write a function C(x) that represents the total cost of the order, where x is the

number of team members. Write a function R(x) that represents the cost remaining and N(x) that

represents the number of team members who pay for shoes. Then find (𝑹

𝑵)(x) and explain what this

function represents. Finally, if there are 11 members on the basketball team, how much does each of

the paying members pay for shoes?

Find C(x).

Find R(x).

Find N(x).

x f(x) g(x)

-3 2 -3

1 1 2

0 5 5

5 7 6

9 4 -1

Find (𝑅

𝑁)(x).

This function represents:

Evaluate (𝑅

𝑁)(x) when x = 11.

Each paying member will pay ___________ for shoes.

Example: For a given triangle, the length of the base is represented by 𝒃(𝒙) = 𝟐𝒙 + 𝟏 and the height is

represented by 𝒉(𝒙) = 𝟓𝒙. Write a function A(x) for the area of the triangle.

Homework Exercises

Let f(x) = 2x + 1 and g(x) = x – 3. State the domain if there are any restrictions.

1. Find (f + g)(x). 2. Find (f – g)(x).

3. Find (f ⋅g)(x). 4. Find (𝑓

𝑔)(x).

Let f(x) = 8𝒙𝟐 and g(x) = 𝟏

𝒙𝟐. State the domain if there are any restrictions.

5. Find (f + g)(x). 6. Find (f – g)(x).

7. Find (f ⋅g)(x). 8. Find (𝑓

𝑔)(x).

Let f(x) = 𝒙𝟐 + 7x + 12 and g(x) = 𝒙𝟐 – 9. State the domain if there are any restrictions.

9. Find (f + g)(x). 10. Find (2f – 3g)(x).

11. Find (f ⋅g)(x). 12. Find (𝑓

𝑔)(-2).

13. Use the table to state the domain of f(x) and g(x). Then find the following:

a. (𝑓 + 𝑔)(1)

b. (𝑓

𝑔) (0)

c. (𝑓

𝑔) (1)

d. 𝑓(−2) − 𝑔(2)

e. (2𝑔 + 𝑓)(−1)

14. Use the graph to state the domain of f(x) and g(x). Then find the following (estimate where necessary):

a. (𝑓 + 𝑔)(1) c. 𝑓(4) − 𝑔(2)

b. (𝑓𝑔)(1) d. (2𝑔 + 𝑓)(3)

15. BUSINESS The function f(x) = 1000 – 0.01𝑥2 models the manufacturing cost per item when x

items are produced, and g(x) = 150 – 0.001𝑥2 models the service cost per item. Write a

function C(x) for the total manufacturing and service cost per item.

16. PROFIT The function f(x) = 4𝑥2 + 2x represents the revenue a company earns x years after

2000, and g(x) = 10x + 125 represents the cost per year. Write a function P(x) for the profit the

company earns per year. (Hint: Profit is the difference of revenue and cost.)

x f(x) g(x)

-2 5 -21

-1 6 -14

0 7 -7

1 8 0

2 9 7

2CP Lesson: Composition of Functions

Perform Compositions of Functions Suppose f and g are functions such that the range of g is a subset of the

domain of f. Then the composite function f ◦ g can be described by the equation [f ° g](x) = f[g(x)].

Example 1: For f = {(1, 2), (3, 3), (2, 4), (4, 1)} and g = {(1, 3), (3, 4), (2, 2), (4, 1)}, find f ◦ g and g ◦ f if they

exist.

f[g(1)] = f(3) = 3 f[g(2)] = f(2) = 4 f[g(3)] = f(4) = 1 f[g(4)] = f(1) = 2,

So f ◦ g = {(1, 3), (2, 4), (3, 1), (4, 2)}

g[f(1)] = g(2) = 2 g[f(2)] = g(4) = 1 g[f(3)] = g(3) = 4 g[f(4)] = g(1) = 3,

So g ◦ f = {(1, 2), (2, 1), (3, 4), (4, 3)}

Example 2: Find [g ◦ h](x) and [h ◦ g](x) for g(x) = 3x – 4 and h(x) = 𝒙𝟐 – 1.

[g ◦ h](x) = g[h(x)] [h ◦ g](x) = h[g(x)]

= g(𝑥2 – 1) = h(3x – 4)

= 3(𝑥2 – 1) – 4 = (3𝑥 − 4)2 – 1

= 3𝑥2 – 7 = 9𝑥2 – 24x + 16 – 1

= 9𝑥2 – 24x + 15

Exercises

For each pair of functions, find f ◦ g and g ◦ f, if they exist.

1. f = {(–1, 2), (5, 6), (0, 9)}, 2. f = {(5, –2), (9, 8), (–4, 3), (0, 4)},

g = {(6, 0), (2, –1), (9, 5)} g = {(3, 7), (–2, 6), (4, –2), (8, 10)}

Find [f ◦ g](x) and [g ◦ f](x), if they exist.

3. f(x) = 2x + 7; g(x) = –5x – 1 4. f(x) = 𝑥2 – 1; g(x) = –4𝑥2

Apply Compositions of Functions Composition of functions can be used in real-world situations when

functions are applied in sequence.

Example: An appliance store is discounting all new dishwashers by 10%. At the same time, the

manufacturer is offering a $100 rebate on all new dishwashers. Danielle is buying a dishwasher that is

priced at $850. Will the final price be lower if the discount is applied before the rebate or if the rebate is

applied before the discount?

First, define variables and functions.

Let x represent the original price of a new dishwasher.

Let f(x) represent the price of a dishwasher after the discount.

Let g(x) represent the price of the dishwasher after the rebate.

Then write equations for f(x) and g(x).

If the discount is applied before the rebate, then the final price of the new dishwasher is represented by

If the rebate is applied before the discount, then the final price of the new dishwasher is represented by

[g ◦ f](850) = _____ and [f ◦ g](850) = ______. So,

Exercises

1. Javier wants to purchase a new television. Electronics Plus offers both an in-store $50 rebate and a 20%

discount on a television that normally sells for $1200. Which provides the better price: taking the discount

before the rebate or taking the discount after the rebate?

2. Corey wants to purchase a new elliptical. A fitness store offers both an in-store $75 rebate and a 5%

discount on an elliptical that normally sells for $2500. Which provides the better price: taking the discount

before the rebate or taking the discount after the rebate?

2CP HW Practice Composition of Functions For each pair of functions, find f ◦ g and g ◦ f, if they exist.

1. f = {(–9, –1), (–1, 0), (3, 4)} 2. f = {(–4, 3), (0, –2), (1, –2)}

g = {(0, –9), (–1, 3), (4, –1)} g = {(–2, 0), (3, 1)}

3. f = {(–4, –5), (0, 3), (1, 6)} 4. f = {(0, –3), (1, –3), (6, 8)}

g = {(6, 1), (–5, 0), (3, –4)} g = {(8, 2), (–3, 0), (–3, 1)}

Find [g ◦ h](x) and [h ◦ g](x), if they exist.

5. g(x) = 3x 6. g(x) = –8x 7. g(x) = x + 6

h(x) = x – 4 h(x) = 2x + 3 h(x) = 3𝑥2

8. g(x) = x + 3 9. g(x) = –2x 10. g(x) = x – 2

h(x) = 2𝑥2 h(x) = 𝑥2 + 3x + 2 h(x) = 3𝑥2 + 1

If f(x) = 𝒙𝟐, g(x) = 5x, and h(x) = x + 4, find each value.

11. f[g(1)] 12. g[h(–2)] 13. h[f(4)]

14. f[h(–9)] 15. h[g(–3)] 16. g[f(8)]

17. g[h(–2)] 18. h[f(5)] 19. f[g(–4)]

20. f[g(–1)] 21. g[h(3)] 22. h[g(7)]

23. [g ◦ (f ◦ h)](–1) 24. [h ◦ (g ◦ f)](0) 25. [f ◦ (h ◦ g)](2)

26. MEASUREMENT The formula f = 𝑛

12 converts inches n to feet f, and m =

𝑓

5280 converts feet to miles m.

Write a composition of functions that converts inches to miles.

2CP WS Inverse Functions The given coordinates are on f(x), find the coordinates for f-1(x)

1. ( - 2 , 4 ) 2. ( 4 , 7 ) 3. ( 0 , 11 ) 4. ( - 3 , - 8 ) 5.( 10, 10 )

Find the algebraic inverse.

6. 115)( xxf 7. 73

1)( xxf 8. 115 xxf

9. 22 xxf 10. 𝑓(𝑥) = √𝑥 − 4

Graph the inverse of the given function.

11. 12.

13. Graph f(x) = x2 + 1 and its inverse.

Restrict the domain of f(x) so that f–1(x) is a function.

14. Graph f(x) = |x – 1| and its inverse.

Restrict the domain of f(x) so that f–1(x) is a function.

15. Show that each of the following functions are inverses by showing that f(g(x)) = x and g(f(x))=x.

a) f(x) = x2 – 4; g(x) = x + 4 b) f(x) = 1

x – 1 ; g(x) =

1x + 1

c) f(x) = 2x + 3; g(x) = x – 3

2 d) f(x) =

2x + 12x – 1

; g(x) = x + 1

2(x – 1)

HW Answers: Operations with Functions 1. 3x – 2

2. x + 4

3. 2𝑥2 − 5𝑥 − 3

4. 2𝑥+1

𝑥−3, x ≠ 3

5. 8𝑥4+1

𝑥2 , x ≠ 0

6. 8𝑥4−1

𝑥2 , x ≠ 0

7. 8, x ≠ 0

8. 8𝑥4, x ≠ 0

9. 2𝑥2 + 7𝑥 + 3

10. 2𝑥2 + 7𝑥 + 3

11. 𝑥4 + 7𝑥3 + 3𝑥2 − 63𝑥 − 108

12. −2

5

13. See below

a. 𝑓(1) + 𝑔(1) = 8 + 0 = 8

b. f(0)/g(0)=7/-7= -1

c. f(1)/g(1)=8/0=undefined

d. 5-(-21)=26

e. 2g(-1)+f(-1)=2(-14)+6= -22

14. See below

a. f(1)+g(1)=0.5+4 = 4.5

b. f(1)*g(1)=0.5(4)=2

c. 2-4= -2

d. 2g(3)+f(3)=2(1.5)+0 = 3

15. 𝐶(𝑥) = 1150 − 0.011𝑥2

16. 𝑃(𝑥) = 4𝑥2 − 8𝑥 − 125

HW Answers: Composition of Functions 1. {(0, –1), (–1, 4), (4, 0)};

{(–9, 3), (–1, –9), (3, –1)}

2. {(–2, –2), (3, –2)};

{(–4, 1), (0, 0), (1, 0)}

3. {(6, 6), (–5, 3), (3, –5)};

{(–4, 0), (0, –4), (1, 1)}

4. does not exist; {(0, 0), (1,

0), (6, 2)}

5. 3x –12; 3x – 4

6. –16x – 24; –16x + 3

7. 3x2 + 6; 3x2 + 36x + 108

8. 2x2 + 3; 2x2 + 12x + 18

9. –2x2 – 6x – 4; 4x2 – 6x + 2

10. 3x2 –1; 3x2 – 12x + 13

11. 25

12. 10

13. 20

14. 25

15. -11

16. 320

17. 10

18. 29

19. 400

20. 25

21. 35

22. 39

23. 45

24. 4

25. 196

26. [m ∘ f ](n) = 𝑛

63,360

HW Answers 2CP WS Inverse Functions

)(for scoordinate thefind , )(on are scoordinategiven The 1 xfxf

1. ( - 2 , 4) Inverse ( 4 , - 2)

2. ( 4 , 7) Inverse ( 7 , 4 )

3. ( 0 ,11) Inverse ( 11, 0)

4. (- 3 ,- 8)Inverse ( - 8, - 3)

5. (10, 10) Inverse (10 ,10)

Find the algebraic inverse.

6. 115)( xxf

yx

yx

yx

xy

15

1

151

115

115

15

11 x

xf

7. 73

1)( xxf

yx

yx

yx

xy

213

3

17

73

1

73

1

2131 xxf

8. 115 xxf

yx

yx

yx

xy

5

11

511

115

115

5

111

x

xf

9. 22 xxf

yx

yx

yx

yx

xy

2

2

2

2

2

2

2

2

21 xxf

10. 4 xxf

yx

yx

yx

yx

xy

4

4

4

4

4

2

2

22

421 xxf

Graph the inverse of the given function.

11.

Function

Points

( - 2 , - 4 )

( 0 , 1 )

( 2 , 6 )

Inverse

Points

( - 4 , - 2 )

( 1 , 0 )

( 6 , 2 )

12. Function

Points

( 4 , 2 )

( 2.5 , -2 )

( - 1, - 4 )

Inverse Points

( 2 , 4 )

( - 2 , 2.5 )

( -4 , -1 )

See graph in class

13. Graph f(x) = x2 + 1 and its inverse. Restrict

the domain of f(x) so that f–1(x) is a function.

Domain restriction of f(x): (−∞, 0)𝑜𝑟 (0, ∞)

14. Graph f(x) = |x – 1| and its inverse.

Restrict the domain of f(x) so that f–1(x) is a function.

Domain restriction of f(x):

(−∞, −1)𝑜𝑟 (−1, ∞)

For #15, see solutions in class