1. INVERSE
FUNCTIONS

2. Recall that to determine by the graph if an equation is a function, we have the vertical line test.
If a vertical line intersects the graph of an equation more than one time, the equation graphed is NOT a function.
This is NOT a function
This is a function
This is a function

3. Horizontal Line Test to see if the inverse is a function.
If the inverse is a function, each y value could only be paired with one x. Let’s look at a couple of graphs.
Look at a y value (for example y = 3)and see if there is only one x value on the graph for it.
For any y value, a horizontal line will only intersection the graph once so will only have one x value
Horizontal Line Test to see if the inverse is a function.

4. This is NOT a one-to-one function This is NOT a one-to-one function
If a horizontal line intersects the graph of an equation more than one time, the equation graphed is NOT a one-to-one function and will NOT have an inverse function.
This is NOT a one-to-one function
This is NOT a one-to-one function
This is a one-to-one function

5. Horizontal Line Test
Used to determine whether a function’s inverse will be a function by seeing if the original function passes the horizontal line test.
If the original function passes the horizontal line test, then its inverse is a function.
If the original function does not pass the horizontal line test, then its inverse is not a function.

6. Steps for Finding the Inverse of a One-to-One Function
y = f -1(x)
Solve for y
Trade x and y places
Replace f(x) with y

7. Let’s check this by doing
Find the inverse of
y = f -1(x) or
Solve for y
Trade x and y places
Yes!
Replace f(x) with y
Ensure f(x) is one to one first. Domain may need to be restricted.

8. Find the inverse of a function :
Example 1: y = 6x - 12
Step 1: Switch x and y:
x = 6y - 12
Step 2: Solve for y:

9. Given the function : y = 3x2 + 2 find the inverse:
Example 2:
Given the function : y = 3x find the inverse:
Step 1: Switch x and y:
x = 3y2 + 2
Step 2: Solve for y:

10. Ex: Find an inverse of y = -3x+6.
Steps: -switch x & y
-solve for y
y = -3x+6
x = -3y+6
x-6 = -3y

11. Finding the Inverse
Try

12. Consider the graph of the function The inverse function is

13. Consider the graph of the function The inverse function isx x x x
The inverse function is
An inverse function is just a rearrangement with x and y swapped.
So the graphs just swap x and y!

14. What else do you notice about the graphs?x x
is a reflection of in the line y = x
The function and its inverse must meet on y = x

15. Graph f(x) and f -1(x) on the same graph.-4 5 5 -4 3 3 4 5 5 4

16. Graph f(x) and f -1(x) on the same graph.-4 -4 -4 -4 -3 -5 -5 -3 4 4

17. Let’s consider the function and compute some values and graph them.
Notice that the x and y values traded places for the function and its inverse.
These functions are reflections of each other about the line y = x
Let’s consider the function and compute some values and graph them.
This means “inverse function”
x f (x)
(2,8)
(8,2)
x f -1(x)
Let’s take the values we got out of the function and put them into the inverse function and plot them
(-8,-2)
(-2,-8)
Is this a function?
Yes
What will “undo” a cube?
A cube root

18. Graph f(x) = 3x − 2 and
using the same set of axes.
Then compare the two graphs.
Determine the domain and range of the function
and its inverse.

19. e.g. On the same axes, sketch the graph of
and its inverse.
Solution: x

20. OR, if you fix the tent in the basement…
Ex: f(x)=2x2-4 Determine whether f -1(x) is a function, then find the inverse equation.
y = 2x2-4
x = 2y2-4
x+4 = 2y2
OR, if you fix the tent in the basement…
f -1(x) is not a function.

21. OR, if you fix the tent in the basement…
Ex: g(x)=2x3
y=2x3
x=2y3
OR, if you fix the tent in the basement…
Inverse is a function!

22. (c) Restricted domain: ( We’ll look at the other possibility
Solution:
(a)
(b) Range of :
(c) Restricted domain:
( We’ll look at the other possibility
in a minute. )
(d) Inverse:
Let
Rearrange:
Swap:
Domain:
Range:

23. Solution:
(b) Range of :
(a)
(c)
Suppose you chose
for the domain
(d) Let
As before
Rearrange:
We now need since

24. Solution:
(a)
Range:
(b)
(c)
Suppose you chose
for the domain
Choosing is easier!
(d) Let
As before
Rearrange:
We now need since
Swap:
Domain:
Range: