Invert a restricted logarithm and carry its range into a cubic

The function $f$ is defined by

$f:x\mapsto\ln(x-a)$, $x\in\mathbb R$, $a<x\leq a+2$,

where $a$ is a positive constant.

(i) Show that $f^{-1}$ exists.2 marks

(ii) Find $f^{-1}(x)$ and state the domain of $f^{-1}$.2 marks

(iii) Find an expression for $g(x)$ for each of the following cases.

(a) $fg(x)=x$,1 mark

(b) $gf(x)=x^2$.1 mark

(iv) The function $h$ is defined by

$h:x\mapsto\dfrac12(x-1)(x+1)(x+5)$, $x\in\mathbb R$.

Find the range of $hf$.2 marks