Minor quibble:

A strictly increasing function is, well, exactly what it sounds like. If x > y, f(x) > f(y). Try to visualize this. It never changes direction, never doubles back on itself. So there's always an inverse, for every output, there's always a unique input.

Strictly increasing guarantees the function is one-to-one. However, it also needs to be onto to guarantee an inverse exists. Of course, you can restrict the codomain to the image of the function, but anyhow...