Warm-up 3.3 Let and perform the indicated operation. 1. 2. 3. 4.

1. 2. 3. 4.

Inverse Functions 3.3B Standard: MM2A5 abcd Essential Question: How do I graph and analyze exponential functions and their inverses?

Vocabulary Inverse relation – A relation that interchanges the input and output value of the original relation Inverse functions – The original relation and its inverse relation whenever both relations are functions nth root of a – b is an nth root of a if bn = a Horizontal Line Test – The inverse of a function f is also a function if and only if no horizontal line intersects the graph of f more than once

Example 1: Graph y = 2x – 4 using a table and a red pencil. Graph y = ½x + 2 using a table and a blue pencil. x y x y -4 -4 -2 1 1 -2 2 2 2 3 3 2 4 4 4 4

-8 -6 -4 -2 2 4 6 8 8 6 4 2 -2 -4 -6 -8

x y x y -4 -4 -2 1 1 -2 2 2 2 3 3 2 4 4 4 4 a. What do you notice about the two tables? The input (x) and output (y) are interchanged

b. With your pencil, draw the line y = x. -8 -6 -4 -2 2 4 6 8 8 6 4 2 -2 -4 -6 -8 y = x b. With your pencil, draw the line y = x.

What is the relationship between the red and blue lines and the line y = x? The line y = x is the line of reflection for the graphs of the red and blue lines. We say that y = 2x – 4 and y = ½x + 2 are inverse functions. Let f(x) = 2x – 4 and f-1(x) = ½x + 2 .

To verify that f(x) = 2x – 4 and are inverse functions you must show that f(f-1(x)) = f-1(f(x)) = x. = f(½x + 2) = 2 (½x + 2) – 4 = x + 4 – 4 = x = f-1(2x – 4) = ½(2x – 4) + 2 = x – 2 + 2 = x

(2). Using composition of functions, determine if f(x) = 3x + 1 and g(x) = ⅓x – 1 are inverse functions? f(g(x)) = f(⅓x – 1) = 3(⅓x – 1) + 1 = x – 3 + 1 = x – 2 NO!

Example 3: Graph y = x2 for x  0 using a table and a red pencil. Graph y = √x using a table and a blue pencil. x y x y 1 1 1 1 2 4 4 2 3 9 9 3 4 16 16 4

-8 -6 -4 -2 2 4 6 8 8 6 4 2 -2 -4 -6 -8

a. What do you notice about the two tables? The input (x) and output (y) are interchanged x y x y 1 1 1 1 2 4 4 2 3 9 9 3 4 16 16 4

b. With your pencil, draw the line y = x. -8 -6 -4 -2 2 4 6 8 8 6 4 2 -2 -4 -6 -8 b. With your pencil, draw the line y = x.

What is the relationship between the red and blue graphs and the line y = x? The line y = x is the line of reflection for the graphs of the red and blue graphs. We say that y = x2 and y = √x are inverse functions. Let f(x) = x2 and f-1(x) = √x .

To verify that f(x) = x2 and are inverse functions you must show that f(f-1(x)) = f-1(f(x)) = x. = = x = = x

To find the inverse of a function that is one-to one, interchange x with y and y with x, then solve for y. Find the inverse: 4. y = 3x + 5 Inverse: x = 3y + 5 x – 5 = 3y

Find the inverse: 5. y = 2x2 – 1 Inverse: x = 2y2 – 1 x + 1 = 2y2

Note: The function y = x2 is not a one-to-one function. If the input and output were interchanged, the graph of the new relation would NOT be a y = x2 x = y2 So, we must restrict the domain of functions that are not one-to-one in order to create a function with an inverse!

Sketch the graph of the inverse of each function. 6). x y 6 3 2 x y -4 -3 2

Sketch the graph of the inverse of each function. 7). x y -2 3 2 x y 6 -5 2