6.4 - Use Inverse Functions Page 196 1 5 9 omit 13 g(f(3)) = -25, f(g(1)) = 1, f(f(0)) = 4 17 30 21 25 f•(f)(x):1/x2 f(f)(x): x 29 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions 4/13/2019 6:15 PM

section 6.4 Revised ©2013, vdang@houstonisd.org Inverse Functions section 6.4 Revised ©2013, vdang@houstonisd.org 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Definitions The result of exchanging the input and output value of a relation is an Inverse Function An inverse “undoes” the function. It switches (x, y) to (y, x) Interchange the x and the y. (make y x and make x y) Resolve for y. You may need to do the “SAT trick” becomes Written in function notation as f –1(x) 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 1 Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)} …to find the inverse, switch the x’s and y’s {(0, –3), (2, 1), (6, 3)} {( , ), ( , ), ( , )} –3 2 1 6 3 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 1 inverse Mirrored Image y = x 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

In f’(x), we have the HORIZONTAL LINE TEST Question If a function is a relation, is an inverse a function as well? inverse NO REPEATING Y’s In f’(x), we have the HORIZONTAL LINE TEST 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 2 Determine the inverse of y = 3x – 2 …to find the inverse, switch the x’s and y’s y = 3x – 2 y = 3 – 2 x x = 3y – 2 x + 2 = 3y 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 3 Determine the inverse of y = (1/4)x – (1/2) 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Your Turn Determine the inverse of y = 5x + 1/3 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 3 Determine the inverse of y = 4x2 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 4 Determine the inverse of y = 3x2 – 5 if x > 0 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Your Turn Determine the inverse of y = x2 + 2 if x < 0 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 4 Determine the inverse of f(x) = 2x3 + 1. and determine whether the inverse is a function. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 4 Determine the inverse of f(x) = 2x3 + 1. and determine whether the inverse is a function. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 5 Determine the inverse of and determine whether the inverse is a function 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 6 Determine the inverse of f(x) = x2 – 2 and determine whether the inverse is a function 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Your Turn Determine the inverse of and determine whether the inverse is a function 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

Verifying Using Compositions Composite f(g(x)). Take the g(x) function and substitute this into the f-function and simplify. For g(f(x)) take f(x) function and substitute this into the g-function and simplify. Notation f(g(x)) is also and (g(f(x)) is If 2 functions are inverses then f(g(x)) = x and g(f(x)) = x. Both equations have to equal x. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 6 Prove that and are inverses through a composition. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Example 7 Prove that and are inverses through a composition. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Your Turn Prove that and are inverses through a composition. 6.4 - Use Inverse Functions 4/13/2019 6:15 PM

6.4 - Use Inverse Functions Assignment Pg 442: 3-11 odd, 15-21 odd, 23-43 EOO (do not graph) 6.4 - Use Inverse Functions 4/13/2019 6:15 PM