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...

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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.