`y = ln(x - 2)^2` Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

We are asked to determine if the function `y=ln(x-2)^2 ` has an inverse by finding out if the function is strictly monotonic on its domain by using the derivative. The domain is `RR-{2} `, in other words, x can be any real number except for x=2.

` y'=(2(x-2))/(x-2)^2=2/(x-2)`

For x<2...

Unlock
This Answer Now

Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.

Start your 48-Hour Free Trial

We are asked to determine if the function `y=ln(x-2)^2 ` has an inverse by finding out if the function is strictly monotonic on its domain by using the derivative. The domain is `RR-{2} `, in other words, x can be any real number except for x=2.

` y'=(2(x-2))/(x-2)^2=2/(x-2)`

For x<2 y'<0 and for x>2 y'>0 so the function is not monotonic and thus it does not have an inverse function.

We can verify this by noting that the graph fails the horizontal line test.

Approved by eNotes Editorial Team