Is this definition correct for the inverse of a function?

Question

Is this definition correct for the inverse of a function?

Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the inverse of $f$ by $f^{-1}$.

Answer 1

The formulation of the definition in the OP assumes existence and uniqueness of the inverse, and existence is not there always while uniqueness has to proved. I would reformulate it as:

Let $\,f:X\to Y\,$ be a function. The function $\,g:Y\to X\,$ is said to be an inverse of $\,f$ if $\,g\circ f=i_X\,$ and $\,f\circ g=i_Y$. We denote the inverse of $\,f\,$ by $\,f^{-1}$.

Answer 2

We only define $f^{-1}$ if $f$ is bijective. Also, as it is written, it doesn't make much sense - what is $g$? (unless you're implying that $g = f^{-1}$)

Other than this, that's pretty much it.