Objectives: 1)Students will be able to find the inverse of a function or relation. 2)Students will be able to determine whether two functions or relations are inverses.

Inverses and Relations A relation is a set of ordered pairs The INVERSE relation, is the set of ordered pairs obtained by reversing the coordinates of each original ordered pair. The domain of a relation becomes the range of the inverse, and the range of a relation becomes the domain of the inverse.

Inverse Relations Two relations are inverse relations IFF (if and only if) whenever one relation contains the element (a, b), the other relation contains the element (b, a). Example:

Example 1 The ordered pairs of the relation {(2, 1), (5, 1) and (2, -4)} are the coordinates of the vertices of a right triangle. Find the inverse of this relation and determine whether the resulting ordered pairs are also the vertices of a right triangle.

You Try It… Find the inverse of each relation: a) b)

Property of Inverse Functions - We can write the inverse of a function as - Suppose and are inverse functions. Then, if and only if

Example 2 Consider the inverse functions: Evaluate

When the inverse of a function is a function, then the original function is said to be one-to-one. To determine if the inverse of a function is a function, you can use the horizontal line test.

How do we find the inverse of a function? Example 3: Find the inverse of 1)Replace f(x) with y in the original equation. 1)Interchange x and y. 2)Solve for y. 3)Replace with f -1 (x). To verify graphically, graph both the function and its inverse. They should be reflected over the line y = x.

You Try It… Find the inverse of each function: a) b)

Composition of Functions In a composition, a function is performed, and then a second function is performed on the result of the first function. The composition of f and g is denoted by

Example 3

You Try It… Find and given

Two functions f and g are inverse functions if and only if both of their compositions are the identity function.

Example 4 Determine whether and are inverses.

You Try It… Determine whether the pair of functions are inverses.