1. Find: ℎ(𝑔 𝑥 ) 𝑔(𝑓 𝑥 ) 𝑓(𝑔 𝑥 ) 𝑔(𝑔 𝑥 ) 𝑓 𝑔 ℎ 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

2. 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

3. Functions
Imagine functions are like the dye you use to color eggs. The white egg (x) is put in the function blue dye, B(x), and the result is a blue egg (y).

4. The Inverse Function “undoes” what the function does.
The Inverse Function of the Blue dye is bleach.
The bleach will “undye” the blue egg and make it white.

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
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

6. 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

7. 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.

8. 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

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

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

11. 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

12. 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)

13. 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
Where is the line of reflection?

14. 𝒚=𝒙
𝒚 = 𝒇(𝒙)
The graph of a function and its inverse are mirror images over the line
𝒚 = 𝒇 −𝟏 (𝒙)
y = x

15. 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

16. 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.

18. No!
Yes!
No!
Yes!

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

20. On the same axes, sketch the graph of
and its inverse.
Notice
Solution:
Using the translation of what is the equation of the inverse function?

21. Domain and Range
The previous example used
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 is the reflection of in the line y = x.
At every point, the x and y coordinates of
switch to become the x and y coordinates of
The values of the domain and range of
swap to become the values of the domain and range of

24. 𝒇 𝒙 =− 𝒙+𝟒 𝟐 −𝟕 State the domain and range of 𝑓(𝑥).
𝒇 𝒙 =− 𝒙+𝟒 𝟐 −𝟕
State the domain and range of 𝑓(𝑥).
Is 𝑓 𝑥 one-to-one? State your reason and the implication of a “yes” or “ no” answer.
Find the equation for 𝑓(𝑥) −1 . Restrict the domain if necessary. Make sure to state the restricted domain.
State the domain and range of 𝑓(𝑥) −1
Graph 𝑓 𝑥 and 𝑓(𝑥) −1 on the same grid.
Show 𝒇 𝒇 −𝟏 𝒙 = 𝒇 −𝟏 (𝒇 𝒙 )=𝒙

25. Homework
Page 149 #1-27 odds, 30

26. Quiz Results
6th Period Average: 94.9% Median: 31 = 96.9% 7th Period Average: 94.2% 8th Period Average: 91.1% Median: 30 = 93.8%