Hey grandunification.

Hint: Check to see if your function is monotonic increasing (or decreasing) in the region of your function to see whether you have a 1-1 correspondence (also assume continuity and differentiation for the mapped transform).

The idea behind the above is that if something is monotonic and can be differentiated (with non-zero derivative at all points), then by the inverse function theorem an inverse function exists across the entire domain and co-domain and thus you have 1-1 properties.