1. Suppose that and find and
Warm Up
Suppose f x ( ) =
and g + 1 2
. Find o . 1 2
Suppose that and
find and

2. Suppose f x ( ) =
and g + 1 2
. Find o .

3. Suppose that and find

4. Suppose that and find

5. Operations on Functions
REVIEW
Perform the indicated operation.
Operations on Functions
Addition:
Subtraction:
Multiplication:
Division:
Composition:

6. Inverse Functions
Section 7.8 in text

7. Many (not ALL) actions are reversible
That is, they undo or cancel each other
A closed door can be opened
An open door can be closed
$100 can be withdrawn from a savings account
$100 can be deposited into a savings account

8. NOT all actions are reversible
Some actions can not be undone
Explosions
Weather

9. Mathematically, this basic concept of reversing a calculation and arriving at an original result is associated with an INVERSE.

10. Actions and their inverses occur in everyday life
Climbing up a ladder
Inverse: Climbing down a ladder
Opening the door and turning on the lights
Inverse: Turning off the lights and closing the door

11. A person opens a car door, gets in, and starts the engine.
Inverse: A person stops the engine, gets out, and closes the car door.

12. Inverse operations can be described using functions.
Multiply x by 5
Inverse: Divide x by 5
Divide x by 20 and add 10
Inverse: Subtract 10 from x and multiply by 20
Multiply x by -2 and add 3
Inverse: Subtract 3 from x and divide by -2

13. Notation
To emphasize that a function is an inverse of said function, we use the same function name with a special notation.
Function, f(x)
Inverse Function of f(x) = f -1 (x)

14. As we noted earlier, not every function has an inverse
As we noted earlier, not every function has an inverse. So when does a function have an inverse?
In words: Each different input produces its own different output.
Graphically: Use the Horizontal Line test.

15. Chapter 7: Polynomial Functions
Line Tests
Vertical Line Test on f:
determines if f is a function
f Function
f Not a Function
Horizontal Line Test on f:
determines if f -1 is a function
Glencoe – Algebra 2
Chapter 7: Polynomial Functions

16. Concept 1

17. Interchange x and y and solve for the new y to obtain f-1(x)
How about if the function is given numerically or symbolically,how do you determine its inverse?
Symbolically:
Interchange x and y and solve for the new y to obtain f-1(x)
Numerically: interchange domain (x) and range ( f(x) )
Interchange
Solve

18. Putting It All Together with Examples
Does this table represent a function?
Does this function have an inverse?
Find the inverse x
f(x) -1 -2 1 2 3 4 -6

19. Example Does this table represent a function?
f(x) -3 10 -2 6 -1 4 1 2 3
-10
Does this table represent a function?
Does this function have an inverse?
Find the inverse

20. Example f(x) =3x +5 f(x) = x3 + 1
Does this equation represent a function?
How do you know?
Does this function have an inverse?
Find the inverse
Confirm the inverse
f(x) = x3 + 1
Does this equation represent a function?
How do you know?
Does this function have an inverse?
Find the inverse
Confirm the inverse