2.6 Inverse Functions

Given the function from the set to the set can be written as: By interchanging the first and second coordinates of each of these ordered pairs, you form the inverse function of f, which is denoted by

Think about the domain and range of the inverse function, compared to that of the original function.

Example Find the inverse of

Let’s look at the previous example and find: Something special happens when we take the composition of the function and its inverse. Let’s look at the previous example and find:

Definition of the Inverse of a Function Let f and g be two functions such that Under these conditions, the function g is the inverse of the function f. The function g is denoted by , and the range of f must be equal to the domain of .

Which of the functions is the inverse of

The Graph of the Inverse of a Function The graph of a function and its inverse are related to each other… let’s try to figure it out! If the point (a, b) is on the function, then what point is DEFINITELY on the inverse?

What geometric transformation does that happen on What geometric transformation does that happen on? **If you don’t remember try to look at the graph of an easy function and its inverse**

Horizontal Line Test for Inverse Functions A function f has an inverse function if and only if no horizontal line intersects the graph of f at more than one point.

Check to see that the inverse of the function will be a function.

Finding the Inverse of a Function Algebraically Use the Horizontal Line Test to decide whether f has an inverse. In the equation for f(x), replace f(x) by y. Interchange the roles of x and y, and solve for y. Replace y by In the new equation. Verify that f and are inverses of each other by showing that the domain of f is equal to the range of , the range of f is equal to the domain of , and

Let’s find the inverse functions from the functions we just looked at

Find the inverse of the following functions

And of course there are some applications  Pg 241-242 #80, 81, 83, 85