Question

Which is the inverse of the function f(x) = x2 − 4?

Answer 1

The function is not one-to-one function, hence the inverse doesn't exist for the whole domain of the function. But if we restrict the domain to $x\in(-\infty,0]$ or $x\in[0,\infty)$, then we can give the inverse.

$f(x)=x^2-4\\\\y=x^2-4\\\\x^2=y+4\\\\x=\sqrt{y+4} \vee x=-\sqrt{y+4}\\\\f^{-1}(x)=\sqrt{x+4}\vee f^{-1}(x)=-\sqrt{x+4}$

So, when $D_{f(x)}:x\in(-\infty,0]$, the inverse is $f^{-1}(x)=-\sqrt{x+4}$ and when $D_{f(x)}:x\in[0,\infty)$, the inverse is $f^{-1}(x)=\sqrt{x+4}$.