Lesson 5.3Inverse Functions

Definition of Inverse Functions

Given a function, f(x), g(x) is an inverse of f(x) if and only if

f(g(x)) = x and g(f(x)) = x

for each domain of f and g

The inverse function of f(x) can be written f--1(x).

Note: The superscript -1 by a function does not mean the reciprocal as it does by a numerical expression.

Inverse Properties

1) If g is an inverse of f, then f is an inverse of g

2) The domain of f is the range of f-1; the range of f is the domain of f-1

3) A function may not have an inverse, but if one exists it is unique.

4) A function will have an inverse if it is one-to-one (Horizontal Line Test)

5) If a function is strictly increasing/decreasing (monotonic) then it is one-to-one

6) If f is continuous, then f-1 is continuous

7) If f is increasing (decreasing), then f-1 is increasing (decreasing)

8) If f is differentiable and f’ does not equal 0, then f-1 is differentiable

The Derivative of an Inverse Function

If f(x) has an inverse, g(x), then

xgfxg

1