2. Copyright © 2011 Pearson Education, Inc. Inverse Functions Section 2.5 Functions and Graphs

3. 2.5 Copyright © 2011 Pearson Education, Inc. Slide 2-3 Definition: One-to-One Function If a function has no two ordered pairs with different first coordinates and the same second coordinate, then the function is called one-to-one. Horizontal Line Test If each horizontal line crosses the graph of a function at no more than one point, then the function is one-to-one. One-to-One Functions

4. 2.5 Copyright © 2011 Pearson Education, Inc. Slide 2-4 Definition: Inverse Function The inverse of a one-to-one function f is the function f -1 (read “f inverse”), where the ordered pairs of f -1 are obtained by interchanging the coordinates in each ordered pair of f. A function is a set of ordered pairs in which no two ordered pairs have the same first coordinates and different second coordinates. If the original function is one-to-one, then the set obtained by interchanging the coordinates in each ordered pair is a function—the inverse function. If a function is one-to-one, then it has an inverse function or it is invertible. Inverse Functions

5. 2.5 Copyright © 2011 Pearson Education, Inc. Slide 2-5 If a point (a, b) is on the graph of an invertible function f, then (b, a) is on the graph of f –1. Since the points (a, b) and (b, a) are symmetric with respect to the line y = x, the graph of f –1 is a reflection of f with respect to the line y = x. Graphs of f and f –1

6. 2.5 Copyright © 2011 Pearson Education, Inc. Slide 2-6 Procedure: Finding f -1 (x) by the Switch-and-Solve Method Do the following to find the inverse of a one-to-one function given in function notation: 1. Replace f(x) by y. 2. Interchange x and y. 3. Solve the equation for y. 4. Replace y by f –1 (x). 5. Check that the domain of f is the range of f –1 and the range of f is the domain of f –1. The Switch-and-Solve Method

7. 2.5 Copyright © 2011 Pearson Education, Inc. Slide 2-7 Theorem: Verifying Whether f and g Are Inverses The functions f and g are inverses of each other if and only if 1. g(f(x)) = x for every x in the domain of f and 2. f(g(x)) = x for every x in the domain of g. The Switch-and-Solve Method