Injectivity equivalent to having a left inverse

[Mr Davis 97]

I know that one can easily prove the result that a function is injective if and only if that function has a left inverse. But is there intuitive reason for this? Same goes for the fact that having a right inverse is equivalent to being surjective. Why are the properties of injectivity and sujectivity related to inverses?

 

[andrewkirk]

When a left inverse is applied to a function, as in $f^{-1}\circ f(x)$, it 'undoes' the effect of the function because, under the right-to-left rule for function composition, $f^{-1}$ is applied after $f$. If the function was not injective $f^{-1}$ could not map back to the original $x$ because there would be more than one possibility.

When a right inverse is applied, as in $f\circ f^{-1}(y)$, the right inverse $f^{-1}$ will not be defined on all elements of its domain - which is the range of $f$ - if $f$ is surjective.