Q.Given $f(x)=x^2-1$ and $g(x)=x+2$, find $(f\circ g)(x)$ and $(g\circ f)(x)$, and show they are not equal.

The inverse $f^{-1}$ undoes a bijective function $f$: $f(x)=y\Leftrightarrow f^{-1}(y)=x$.

A function $f:A\to B$ has an inverse only if it is bijective (both one-one and onto). The inverse function $f^{-1}:B\to A$ is defined by

$f^{-1}(y)=x \iff f(x)=y$

so it maps each element of the codomain back to the unique element of the domain that produced it. Consequently $f^{-1}\big(f(x)\big)=x$ for all $x\in A$ and $f\big(f^{-1}(y)\big)=y$ for all $y\in B$.

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