y = e^(ln(3x)) Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function.

We are asked to determine if the function y=e^(ln(3x)) has an inverse function by finding if the function is strictly monotonic on its entire domain using the derivative. The domain of the function is x>0.
First use the properties of the exponential and logarithm functions to simplify the function:
y=e^(ln(3x))=3x so y'=3. Thus the function is strictly monotonic and has an inverse function.
The graph:

(Note: if you did not see to simplify the right hand side, we can get the derivative:
y'=3/(3x)e^(ln(3x))=1/xe^(ln(3x)) . Now e^u>0 for all real inputs, and 1/x>0 for x>0 (the domain of the function) so we still get the function being strictly monotonic.)