An absolute-value restriction and a composite range

Functions $f$, $g$ and $h$ are defined by

$f:x\mapsto|x^2-6x+5|$ for $x\in\mathbb R$, $2\leq x\leq6$,

$g:x\mapsto\ln(x^2)$ for $x\in\mathbb R$, $x\ne0$,

$h:x\mapsto\dfrac1{2-x}$ for $x\in\mathbb R$, $x\leq\dfrac32$.

(i) Give a reason why $f$ does not have an inverse.

(ii) State the longest possible interval in the domain of $f$, to which the restriction of $f$ would have an inverse. Hence, define $f^{-1}$ in similar form for function $f$ restricted to this interval.

(iii) Show that the composite function $gh$ exists.

(iv) Determine the exact range of function $gh$.