How to find the inverse of a 2-part function

Question

I got the function $$f(x)=\begin{cases}{x^2+2,\text{if }x\in(-2, 0]}\\{3x-7,\text{if }x\in(0, 3)}\end{cases}$$ and I want to find the $f^{-1}(x)$. How to do that? I find the inverse of the $2$ parts separate or somethin else?

Answer 1

For the first part. Write $y=x^2+2$. Then $y\in [2,6)$. Now swap $x$ and $y$. Then you get $x=y^2+2$ so $y=\pm \sqrt{x-2}$ and $x\in [2,6)$. Choose part with $-$ because $x<0$ (in OP). Do the second part your self.

$$f^{-1}(x)=\begin{cases}{-\sqrt{x-2},\text{if }x\in[2,6)}\\{{x+7\over 3},\text{if }x\in(-7, 2)}\end{cases}$$