5.3 Inverse Functions

Definition of Inverse Function A function of “g” is the inverse function of the function “f” if: f(g(x)) = x for each x in the domain of g and g(f(x)) = x for each x in the domain of g The function g is denoted by f -1 (read “f inverse”)

Thm: Reflective Property of Inverse Functions: The graph of “f” contains the point (a, b) if and only if the graph of f -1 contains the point (b, a). Example: If f contains the points: {(0, 1), (1, 2), (2, 3), (3, 4), (4, 5)} Find f -1 : {(, ), (, ), (, ), (, ), (, )} Graph to verify that they are inverses.

Example: Show that f and g are inverses analytically and graphically: f(x) = 5x + 1 and g(x) = 1/5x – 1/5

Thm: The Existence of an Inverse 1. A function has an inverse function if and only if it is one-to-one. “HORIZONTAL LINE TEST” 2. If “f” is strictly monotonic on its domain, then it is one-to-one and therefore has an inverse function. Strictly Monotonic  if the function is either increasing or decreasing on its entire domain.

Continuity and Differentiability of Inverse Functions Let “f” be a function whose domain is on an Interval “I”. If “f” has an inverse function, then the following statements are true: 1. If “f” is continuous on its domain, then f -1 is continuous of its domain. 2. If “f” is increasing on its domain, then f -1 is increasing of its domain. 3. If “f” is decreasing on its domain, then f -1 is decreasing of its domain. 4. If “f” is differentiable at c and f’(c) = 0, then f -1 is differentiable at f(c).

The Derivative of an Inverse Function Let “f” be a function that is differentiable on an Interval “I”. If “f” has an inverse function “g”, then g is differentiable at any x for which f’(g(x)) = 0. Moreover, g’(x) = 1 f’(g(x)) f’(g(x)) = 0

Example: Let f(x) = x a. What is the value of f -1 (x) when x = 9? b. What is the value of (f -1 )’(x) when x = 9?

Example: Let f(x) = x 2 + 5x + 6 a. What is the value of f -1 (x) when x = 12? b. What is the value of (f -1 )’(x) when x = 12?