1. Lesson 5.3 Inverse Functions

2. 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.

3. Inverse Properties
If g is an inverse of f, then f is an inverse of g
The domain of f is the range of f-1; the range of f is the domain of f-1
A function may not have an inverse, but if one exists it is unique.
A function will have an inverse if it is one-to-one (Horizontal Line Test)
If a function is strictly increasing/decreasing (monotonic) then it is one-to-one
If f is continuous, then f-1 is continuous
If f is increasing (decreasing), then f-1 is increasing (decreasing)
If f is differentiable and f’ does not equal 0, then f-1 is differentiable

8. The Derivative of an Inverse Function
If f(x) has an inverse, g(x), then

9. Dy/dx=1/dy/dx