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

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

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.

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

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)

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?

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

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!

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

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 .

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

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%