Lesson 5-5: Inverse Functions and Relations Advanced Math Topics

Property of Inverse Functions Two functions f and g for which: f(a) = b and g(b) = a * these can also be written as f(a) = b and f -1(b) = a

To write the equation of the inverse for each function: Replace f(x) with y. Switch x and y. Solve for y. Replace y with f -1(x).

Example #1- would be f(x) = x - 5 y = x – 5 x = y – 5 x + = y + 5 +5 x + 5 = y f -1(x) = x + x x

Graphing functions and their inverses when (a,b) is on the graph of a function, then __________ is on the graph of the inverse The ___________ is the reflection of the graph over the line ________. (b, a) inverse y = x

An inverse _____________________ An inverse _____________________. On a graph you can use a _______________ to determine whether a _____________ is also a function. Horizontal Line Test: If a ____________ crosses the graph more than once, then the function’s inverse _______________. may or may not be a function Horizontal line test inverse horizontal line does not exist *** do the horizontal line test on the original function

1. Make an x/y chart for f(x) graph the points and connect the dots To Graph a function and it’s inverse (put this on the blank space at the bottom of the page) 1. Make an x/y chart for f(x) graph the points and connect the dots 2. Make an x/y chart for f-1(x) by switching the x and y coordinates and then graph and connect the dots

Example 1: Find and Graph an inverse function Find the inverse of Graph f(x) and f-1(x)

Examples Cont… graph the function and its inverse

Inverse Functions Two functions f and g are inverse functions if and only if both of their compositions are the identity function. [f o g](x) = x and [g o f](x) = x So, if you plug g into f and simplify you get x and if you plug f into g and simplify you get x BOTH have to give x for functions to be inverses

Determine if the functions are inverses

Determine if the functions are inverses