1. Inverse Functions
Lesson 8.2

2. Definition
A function is a set of ordered pairs with no two first elements alike.
f(x) = { (x,y) : (3, 2), (1, 4), (7, 6), (9,12) }
But ... what if we reverse the order of the pairs?
This is also a function ... it is the inverse function
f -1(x) = { (x,y) : (2, 3), (4, 1), (6, 7), (12, 9) }

3. Example
Consider an element of an electrical circuit which increases its resistance as a function of temperature.
T = Temp
R = Resistance
-20 50
150 20
250 40
350
R = f(T)

4. Now we would say that g(R) and f(T) are inverse functions
Example
We could also take the view that we wish to determine T, temperature as a function of R, resistance.
R = Resistance
T = Temp 50
-20
150
250 20
350 40
T = g(R)
Now we would say that g(R) and f(T) are inverse functions

5. Terminology If R = f(T) ... resistance is a function of temperature,
Then T = f-1(R) ... temperature is the inverse function of resistance.
f-1(R) is read "f-inverse of R“
is not an exponent
it does not mean reciprocal

6. Does This Have An Inverse?
Given the function at the right
Can it have an inverse?
Why or Why Not? x Y 1 5 2 9 4 6 7
NO … when we reverse the ordered pairs, the result is Not a function.

7. Finding the Inverse
Try

8. Composition of Inverse Functions
Consider
f(3) = 27   and   f -1(27) = 3
Thus, f(f -1(27)) = 27
and f -1(f(3)) = 3
In general   f(f -1(n)) = n   and f -1(f(n)) = n (assuming both f and f -1 are defined for n)

9. Graphs of Inverses Again, consider
Set your calculator for the functions shown
Dotted style
Use Standard Zoom
Then use Square Zoom

10. Graphs of Inverses
Note the two graphs are symmetric about the line y = x

11. Investigating Inverse Functions
Consider
Demonstrate that these are inverse functions
What happens with   f(g(x))?
What happens with  g(f(x))?
Define these functions on your calculator and try them out

12. Domain and Range The domain of f is the range of f -1
The range of f is the domain of f -1
Thus ... we may be required to restrict the domain of f so that f -1 is a function

13. Domain and Range Consider the function h(x) = x2 - 9
Determine the inverse function
Problem =>  f -1(x) is not a function

14. Inverse Pumpkins
In a recent pumpkin launching contest, one launcher misfired so that the pumpkin went straight up into the air (!!) and came back down to land on the launch personnel!  
Below is the graph of the height of the pumpkin as a function of time h(t) 
Note ... the curve is not the path of the pumpkin. 

15. Inverse Pumpkins What is the "hang time" of the launch?
Restrict the domain of h(t) so that it has an inverse that is a function
Graph the inverse of this function
Change the story to go with your new graph.  Explain in your story why it makes sense that the inverse is a function.

16. Assignment
Lesson 8.2
Page 370
Exercises 1 – 49 EOO