Composite and Inverse Functions

Lesson 2.4

2

Composition of Functions

• Consider two functions where the output of one is the input of the next

• Example Square yds/hr mowed is a

function of how fast you push the mowerA = f(s)

The time required to mow is a function of square yds/hr you cover T = g(A)

speed

sq yds/hr

f(s)

Time

g(A)

3

Composition of Functions

Given the following functions

• Q = f(p) The number of barrels of oil sold when the price is p dollars per barrel

• R(Q) is the revenue earned when Q barrels are sold

• What is R(f(p)) ? What are the units of each function?

4

Composition of Functions

• Given

• Find the following compositions

2( ) 1 ( ) 2 3f x x g x x

( (1))f g

( (2))g f( ( ))g g n

Try using your calculator

Try using your calculator

5

Inverse Functions

• What if we cram a numberup the spout of a function and out of the funnel popsthe number that wouldhave given us the result??

• The function that does this is called theinverse function

1

-3

4

3

3

1

Use spreadsheet to evaluate inverse

of a function

Use spreadsheet to evaluate inverse

of a function

6

Perspectives for Input and Output

• Suppose you are told 1 gallon of paint covers 250 ft2

You might derive the function

• It is just as reasonable to consider how many gallons are needed for a certain area

250

( ) 250

Area g

f g g

250

( )250

areaGallons

Ah A

7

Perspectives for Input and Output

• The mathematical relationship is the same The input on one f(g) is the output on h(A)

• We would say the functions have an inverse relationship

250

( ) 250

Area g

f g g

250

( )250

areaGallons

Ah A

8

Inverse Function Notation

• For the inverse of function f, we use the notation f -1

• Note that this is not the same as a negative exponent It is not

1

f

1

( ) 250

( )250

A f g g

Ag f A

9

Finding Inverse Values from a Table

• Given the following table which defines the function f

• Determine f(-2) f -1(2) f -1(-4) f(-1)

x -2 -1 0 1 2

f(x) 6 -4 3 2 9

10

Finding Inverse Values from a Graph

• Write some ordered pairsfor the functiondefined by thisgraph

• Determinef -1(0)f -1(-2)

x f(x)

• Are there multiple answers

• Is the inverse even a function?

• Are there multiple answers

• Is the inverse even a function?

11

Finding the Inverse Formula

• Given the formula

• Find the inverse function f -1(V)

• Strategy Write in formula notation Solve for the independent

variabler = ?

34( )

3V f r r

34

3V r

12

Domain and Range of An Inverse Function

• Note that the domain of the original function becomes the range of the inverse Thus restrictions on the original domain affect

the range of the inverse

• AlsoThe range of the original may be restricted This affects the domain of the inverse

• Consider the inverses of these functions5

( )3

xf x

x

2( ) 25g x x

As we saw on slide 10, some inverses might not

even be functions

As we saw on slide 10, some inverses might not

even be functions

13

Assignment

• Lesson 2.4

• Page 82

• Exercises1 – 37 odd