1. 32𝑥−16 4 𝑥 2 −8𝑥+10 16 𝑥 2 −32𝑥+11 1024 𝑥 2 −192𝑥+11 Find: ℎ(𝑔 𝑥 )
𝒇 𝒙 = 𝒙 𝟐 −𝟐𝒙+𝟑 𝒈 𝒙 =𝟒𝒙−𝟐 𝒉 𝒙 =𝟖𝒙
Find:
ℎ(𝑔 𝑥 )
𝑔(𝑓 𝑥 )
𝑓(𝑔 𝑥 )
𝑔(𝑔 𝑥 )
𝑓 𝑔 ℎ 𝑥
32𝑥−16
4 𝑥 2 −8𝑥+10
16 𝑥 2 −32𝑥+11
16𝑥−10
1024 𝑥 2 −192𝑥+11

2. Quiz Results
5th Period Average: 94% Median: 98% 7th Period Average: 92.9% Median: 94% 8th Period Average: 90.5% Median: 95.2%

3. Section 4-5 Inverse Functions
Objective: To find the inverse of a function, if the inverse exists.
Inverse Definition – Function Composition
Finding the Inverse Algebraically
Graphing the Inverse
Horizontal Line Test: One to one Function
Domain & Range

4. Inverse Functions The inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function for x≥𝟎: f(x) = x2 x
f(x) y
𝒇 −𝟏 (𝒙) 9 3 3 9 9 3 3 9 9 3 3 9 9 3 3 x2 9 9 3 3 9 9 9 3 3 3 9 9

5. In the same way, the inverse of a given function will “undo” what the original function did.
For example, let’s take a look at the square function for x≥𝟎 : f(x) = x2 x y
𝒇 −𝟏 (𝒙)
f(x) 5 25 5 5 5 25 25 5 5 25 25 5 5 x2 25 5 5 25 5 25 25 5 25 5 5 5

6. Inverse Function Definition
The inverse of a function f is written 𝑓 −1 and is read “f inverse”
𝑓 −1 (𝑥) is read, “f inverse of x”
Inverse Function Definition
Two functions f and g are called inverse functions if the following two statements are true:
1. 𝑔(𝑓 𝑥 )= 𝑥 for all x in the domain of f.
2. 𝑓(𝑔 𝑥 )=𝑥 for all x in the domain of g.

7. 𝑔(𝑥)=2𝑥 +1 𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥 =𝑥 Example
Consider the functions f and g listed below. Show that f and g are inverses of each other.
𝑔(𝑥)=2𝑥 +1
Show that:
𝑓 𝑔 𝑥 =𝑔 𝑓 𝑥 =𝑥

8. Example
𝑔(𝑥)=2𝑥 +1
𝑓 𝑔 𝑥 =
𝑓(𝟐𝒙+𝟏)
= 𝟐𝒙+𝟏 −1 2
= 2𝑥 2 =𝑥

9. Example
𝑔(𝑥)=2𝑥 +1
𝑔 𝑥−1 2
𝑔 𝑓 𝑥 =
=2 𝑥−
=𝑥−1+1 =𝑥

10. x = 3y2 + 2 Find the inverse of a function algebraically:
Given the function: f(x) = 3x Find the inverse.
*Note: You can replace f(x) with y.
x = 3y2 + 2
Step 1: Switch x and y
Step 2: Solve for y
𝒇 −𝟏 𝒙 = 𝒙−𝟐 𝟑

11. has an inverse point of (7, 4)
Graphically, the x and y values of a point are switched.
The point (4, 7)
has an inverse point of (7, 4)
AND
The point (-5, 3)
has an inverse point of (3, -5)

12. 𝒚=𝒙 x 1 2 3 4 y 8 16 x 1 2 4 8 16 y 3 Where is the line of reflection?
Graphically, the x and y values of a point are switched.
If the function 𝒈(𝒙) contains the points: x 1 2 3 4 y 8 16
then its inverse 𝒈 −𝟏 (𝒙) contains the points x 1 2 4 8 16 y 3
𝒚=𝒙
Where is the line of reflection?

13. Vertical and Horizontal Line Test
Does the graph pass the vertical line test?
Does the graph pass the horizontal line test?
What does passing/not passing the vertical or horizontal line test mean?
𝒇 𝒙 = 𝟒 – 𝒙𝟐

14. The Vertical Line Test
If the graph of 𝑦 = 𝑓(𝑥) is such that no vertical line intersects the graph in more than one point, then f is a function.

16. No!
Yes!
No!
Yes!
Restrict the Domain

17. 𝒇(𝒙) On the same axes, sketch the graph of and its inverse. Notice
Solution: x

18. What is the equation of the inverse function?
On the same axes, sketch the graph of
and its inverse.
𝒇(𝒙)
Notice
Solution:
What is the equation of the inverse function?

19. 𝒇 𝒙 = 𝒙−𝟐 𝟐
What are the domain and range of the function and of the inverse function?
The Domain of f(x) is
𝑥≥2
The Range of f(x) is
𝑦≥0

20. What do you notice? 𝑥≥2 𝑥≥0 𝑦≥0 𝑦≥2
𝒇 𝒙 = 𝒙−𝟐 𝟐
What are the domain and range of the function and of the inverse function?
The Domain of f(x) is
The Domain of 𝒇 −𝟏 (𝒙) is
𝑥≥2
𝑥≥0
The Range of f(x) is
The Range of 𝒇 −𝟏 (𝒙) is
𝑦≥0
𝑦≥2
What do you notice?

21. Domain and Range The Domain of is Since is found by swapping x and y,
the values of the Domain of give the values of the Range of
Domain
Range

22. Domain and Range
The previous example used
Similarly, the values of the range of
give the values of the domain of
Range
Domain

23. GRAPHING SUMMARY
The graph of 𝒚= 𝒇 −𝟏 (𝒙) is the reflection of 𝒚=𝒇(𝒙) over the line 𝒚=𝒙
At every point, the x and y coordinates of 𝒚=𝒇(𝒙) switch to become the x and y coordinates of 𝒚= 𝒇 −𝟏 (𝒙)
The values of the domain and range of 𝒚=𝒇(𝒙) swap to become the domain and range of 𝒚= 𝒇 −𝟏 (𝒙)

24. Classwork

25. Homework A: Page 149 #1-27 odds, 30 *Hint on 30: 𝒇 𝒙 =𝒎𝒙+𝒃
B: Page 149 #5-29 odds,30,31
*Hint on 29: 𝒇°𝒈=𝒇 𝒈(𝒙 )