1. BellWork

2. INVERSE
FUNCTIONS

3. OBJECTIVE Determine if a function is one-to-one
Find the inverse of a function

4. Remember we talked about functions---taking a set X and mapping into a Set Y1 2 3 4 5 10 8 6 1 2 2 4 3 6 4 8 10 5
Set X
Set Y
An inverse function would reverse that process and map from SetY back into Set X

5. If we map what we get out of the function back, we won’t always have a function going back!!!1 2 2 4 3 6 4 8 5

6. 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

7. 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.

8. 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

9. 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.

10. Does this function have an inverse?

11. Does this function have an inverse?

12. Does this function have an inverse?

13. Does this function have an inverse?

14. Does this function have an inverse?

15. Does this function have an inverse?

16. 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

17. Find the inverse of
y = f -1(x) or
Solve for y
Trade x and y places
Replace f(x) with y
Ensure f(x) is one to one first. Domain may need to be restricted.

18. Find the inverse of a function : Step 1: Replace f(x) with y
Example 1: f(x) = 6x - 12
Step 1: Replace f(x) with y
Step 2: Switch x and y:
x = 6y - 12
Step 3: Solve for y:

19. 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:

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

21. Finding the Inverse