3. a. Explain why the function

defined on the domain

has an inverse.

  1. Suppose the inverse is called g. Evaluate g'(0).

Solution: a

Since f(x) is a cubic its graph is increasing for all x or its graph has a local max and a local min. We check by finding the critical points:

 

Thus f(x) is a cubic that has a local max at x = 1-1/Ö 3, a local min

at x = 1-1/Ö 3, and is strictly decreasing between these numbers.

In other words, f(x) is one-to-one on the specified domain, so has an inverse on it.

b. Starting from g(f(x)) = x and differentiating we get
g'(f(x)) = 1/f'(x). Next, the only point in the restricted domain where f(x) = 0 is x = 1. Consequently,

g'(0)= 1/f'(1) = 1/(-1)= -1