Determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

Question

Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

Answer 1

Step 1:

The function is .

Consider .

Derivative on each side by .

Apply the power rule of derivative :.

Step 2:

To examine the behavior of a function, equate the derivative to zero.

The x values are .

Answer 2

Step 3:

The function is .

The domain of the function is .

There are four regions to examine the behavior of the function.

First region  .

Consider a test point in the region.

The derivative is negative, the function is decreasing over .

Second region  .

Consider a test point in the region.

The derivative is positive, the function is increasing over .

Third region  .

Consider a test point in the region.

The derivative is negative, the function is decreasing over .

Fourth region  .

Consider a test point in the region.

The derivative is positive, the function is increasing over .

A monotonic function is increasing over and .

A monotonic function is decreasing over and .

Therefore the function is not strictly monotonic.

Solution :

The function is not strictly monotonic.