Compose through a restricted range, then use symmetry for area

(a) Functions $f:x\mapsto(x-2)(x-4)$ for $x\in\mathbb R$, $x<4$; $g:x\mapsto x/(x+3)$ for $x\in\mathbb R$, $x\ne-3$.

Determine if $f$ is one-one, justifying your answer.1 mark

Only one of $fg$ and $gf$ exists. Give a definition (including domain) of the composite that exists, and explain why the other does not exist. Find the range of the composite that exists.6 marks

(b) $h:x\mapsto a(1+e^{-x})$, $x\in\mathbb R_0^+$, where $a$ is positive. Find $h^{-1}(x)$ and its domain.3 marks

Given that $h$ and $h^{-1}$ intersect where $x=b$, and $\int_0^b h(x)\, dx=I$, find the area bounded by $h, h^{-1}$ and the axes in terms of $I, b$.2 marks

[2010/TJC/Prelim/II/3]