Does the Definition of an Inverse Depend on Injectivity and Surjectivity?

[Bipolarity]

Consider a function $f(x)$ and its inverse $g(x)$.

Then $(f \circ g)(x) = x$ and $(g \circ f)(x) = x$

Are both these statements separate requirements in order for the inverse to be defined? Is it possible that one of the above statements is true but not the other? If so, could I see an example of such a case?

Otherwise, could you prove one having knowledge of the other?

Thanks!

BiP

 

[mathwonk]

do you know what injective and surjective functions are? every injective g has an f with the first property, but only surjective g's have an f with the second property.

e.g. if g is multiplication by 2, as a map from the integers to themselves, then there is an f with property 1. namely f sends every even integer to half itself, and sends every odd integer to zero.