Prove that if f is strictly increasing on a domain, D, then its inverse f^{-1} is strictly increasing on f(D).
This follows directly from the defintions.
A function is strictly increasing if
$\displaystyle \forall a,b \in D$ if $\displaystyle a < b \iff f(a) < f(b)$
What is the domain of $\displaystyle f^{-1}(x)$?
What is the range of $\displaystyle f^{-1}(x)$?
You just need to show for all $\displaystyle c,d \in f(D)$ if $\displaystyle c <d \iff f^{-1}(c) < f^{-1}(d)$
Can you show the defintion for the inverse holds using the above?