1. Composite and Inverse Functions
Lesson 2.4

2. Composition of Functions
speed
sq yds/hr
f(s)
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 mower A = f(s)
The time required to mow is a function of square yds/hr you cover T = g(A)
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
Try using your calculator

5. Use spreadsheet to evaluate inverse of a function
Inverse Functions
What if we cram a number up the spout of a function and out of the funnel pops the number that would have given us the result??
The function that does this is called the inverse function 1 -3 4 3 3 1
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

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

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

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 1 2
f(x) 6 -4 3 9

10. Finding Inverse Values from a Graph
Write some ordered pairs for the function defined by this graph
Determine f -1(0) f -1(-2) x
f(x)
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 variable 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
Also The range of the original may be restricted
This affects the domain of the inverse
Consider the inverses of these functions
As we saw on slide 10, some inverses might not even be functions

13. Assignment
Lesson 2.4
Page 90
Exercises 1 – 37 Odd