A piecewise inverse and two identity domains

The function $f$ is defined by

$f:x\mapsto\begin{cases}\cos(\pi x/2),&0\leq x<1,\\1-x,&1\leq x\leq2.\end{cases}$

(i) Sketch the graph of $f$, indicating clearly the coordinates of the axial intercepts and endpoints.

(ii) Show that $f$ has an inverse, and define it in similar form.

(iii) Find the range of values of $x$ for which $ff^{-1}(x)=f^{-1}f(x)$.

(iv) The function $g$ is defined by $g:x\mapsto e^x$, $0\leq x\leq\ln2$.

Find the range of $fg$.