Ch. 1 – Functions and Their Graphs 1.6 – Inverse Functions

Inverse Functions Given a function f(x), its inverse function f-1(x) is defined if . Note that f-1 is not the same as the reciprocal of f(x)! Ex: Show that f(x) = 2x – 6 and are inverse functions. f(g(x)) = Since f(g(x)) = x, the two functions are inverses.

Inverse Functions Properties of an inverse function: Its domain will be the range of f(x) Its range will be the domain of f(x) If point (a, b) lies in f, then point (b, a) lies in f-1 The graphs of a function and its inverse are symmetric about the line y = x

Ex: Find the inverse function of . To find an inverse, switch f and x and solve for f! Check your answer by graphing both functions. Can you reflect f across the line y=x to get f-1?

Find f-1(x).

Do all functions have inverses? Ex: Does f(x) = x2 have an inverse? If we switch f and x and solve for f, we get The inverse function is not a function because it fails the vertical line test! Since the inverse function failed the vertical line test, we can use the horizontal line test to determine if a function has an inverse. One-to-one function = each output corresponds to exactly one input For a function to have an inverse, it must be one-to-one

Ex: Find the inverse function of: The domain is restricted, so our inverse function will be restricted as well! Graph this function. What’s its domain and range? Now we’ll solve for the inverse. We can’t have the ± in our answer because f-1 would not be a function then, so we need to choose whether it should be a + or – based on the domain and range… Graph the function to decide the correct answer! Remember, the domain and range flip-flop for f-1

Does f(x) have an inverse? Yes No