Warm Up Chain Reaction Choose one team member to start problem #1. After the first person evaluates 𝑓 𝑥 , pass the paper to the right. The next team member evaluates 𝑔 𝑓 𝑥 . Keep passing the paper to the right until you have a solution. Have a different team member start problem #2. 𝑓 𝑥 =2𝑥 g 𝑥 =𝑥+6 ℎ 𝑥 = −𝑥+10 2 𝑘 𝑥 = 𝑥 2

𝑥 2 −4𝑥+4 𝑘(ℎ(𝑔(𝑓(𝑥)))) 𝑥 2 −4𝑥+1 𝑘(ℎ(𝑔(𝑓 𝑥+1 ))) 𝑘(ℎ(𝑔(𝑓 2− 𝑥 2 ))) Warm Up Chain Reaction Working through the functions in order, find: 𝑘(ℎ(𝑔(𝑓(𝑥)))) 𝑥 2 −4𝑥+4 𝑥 2 −4𝑥+1 𝑘(ℎ(𝑔(𝑓 𝑥+1 ))) 𝑘(ℎ(𝑔(𝑓 2− 𝑥 2 ))) 𝑥 4 𝑘(ℎ(𝑔(𝑓 2𝑥 3 +1 ))) 12 𝑥 2 −4𝑥 3 +1

Quiz Results 2nd Period Average: 94.8% Median: 29 = 96.7% 3rd Period Average: 94.0% 4th Period Average: 92.0% Median: 28.5 = 95%

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

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).

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.

For example, let’s take a look at the square function: f(x) = x2 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: 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

For example, let’s take a look at the square function: f(x) = x2 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: 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

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.

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

Find the inverse of a function algebraically: Given the function: f(x) = 3x2 + 2 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

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

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

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 horizontal line test mean? 𝒇 𝒙 = 𝟒 – 𝒙𝟐

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.

No! Yes! No! Yes!

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

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?

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

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 . Similarly, the values of the range of give the values of the domain of

GRAPHING SUMMARY The graph of is the reflection of in the line y = x. At every point, the x and y coordinates of become the y and x coordinates of . The values of the domain and range of swap to become the values of the range and domain of .

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

Homework Page 149 #1-27 odds, 30