 Solution

Let $u<v$ lie in $f([a,b])$, and put $x=g(u)$ and $y=g(v)$. Equality $x=y$ would imply $u=v$. If $x>y$, strict monotonicity of $f$ would give $u=f(x)>f(y)=v$. Both are impossible, so $x<y$. Therefore the inverse function $g$ is strictly increasing.