A quarter ellipse becomes a self-inverse circle

Sketch the curve $y^2+ax^2=5$ for $x\ge0$, $y\ge0$, where $a$ is positive.1 mark

$f:x\mapsto\sqrt{5-ax^2}$, $x\in\mathbb R$, $0\le x\le\sqrt{5/a}$; $g:x\mapsto1+e^{-x}$, $x\in\mathbb R$, $x\ge0$.

Show that $f^{-1}$ exists and define $f^{-1}$ in a similar form.4 marks

Given $f^2(x)=x$ for all $x\in\mathbb R$, $0\le x\le\sqrt{5/a}$,

(i) show $a=1$ without evaluating $f^2(x)$,2 marks

(ii) using (i), show $fg$ exists and find its corresponding range.3 marks

[2011/ACJC/Prelim/I/9]