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

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

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

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

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

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

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

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

x = 3y2 + 2 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

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)

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?

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

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! Restrict the Domain

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

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?

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

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?

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

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

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)

Classwork

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 )