Inverse of a Function

Definition. For a bijection $f:X\to Y$, the inverse is the function $g:Y\to X$ defined by $g(y)=x$ whenever $f(x)=y$; write $g=f^{-1}$. A function with an inverse is invertible.

Composition characterisation. If $f:X\to Y$ is a bijection with inverse $g:Y\to X$, then $g\circ f=I_X$ and $f\circ g=I_Y$ (the identity functions on $X,Y$). Conversely, if $f:X\to Y$ and $g:Y\to X$ satisfy $g\circ f=I_X$ and $f\circ g=I_Y$, then both $f$ and $g$ are automatically bijections and are inverses of each other ($f^{-1}=g,\ g^{-1}=f$). This gives a second, equivalent definition of invertibility purely via composition, without mentioning "bijection" directly.

$f^{-1}$ is only ever defined for a bijection: if $f$ is not one-to-one, some $y$ has two different pre-images $a\ne b$ with $f(a)=f(b)=y$, so $f^{-1}(y)$ could not be assigned a single value; if $f$ is not onto, some $y$ has no pre-image at all, so $f^{-1}(y)$ has nothing to be.

Example. $A=\{1,2,3,4\},\ f=\{(1,2),(2,4),(3,1),(4,3)\}$ (a bijection $A\to A$): $f^{-1}=\{(2,1),(4,2),(1,3),(3,4)\}$, i.e. $\{(1,3),(2,1),(3,4),(4,2)\}$ once re-sorted.

Working rule for $f:R\to R$. (i) Write $y=f(x)$; (ii) solve algebraically for $x$ in terms of $y$; (iii) that expression is $f^{-1}(y)$; (iv) finally relabel $y$ as $x$ to write $f^{-1}(x)$. …