$\displaystyle f(x)=\left | x \right |$

I did this:

$\displaystyle \left | x \right |= \left\{\begin{matrix}

x; x\in \mathbb{R}^+\\

-x ;x\in \mathbb{R}^-

\end{matrix}\right.$

$\displaystyle \therefore$

$\displaystyle f^{-1}(x):= \left\{\begin{matrix}x;y\in \mathbb{R}^+\\

-x; y\in \mathbb{R}^{-} \end{matrix}\right.$

-but, is there a more 'technical' way of computing this rather than using the definition of both absolute value and inverses such as in the following.

$\displaystyle y=2^x$

$\displaystyle \ln(y)=x\ln(2)$

$\displaystyle \ln(x)=y\ln(2)$

$\displaystyle y=\frac{\ln(x)}{\ln{2}}$

Thank you.

$\displaystyle \int$