The function \(f\) is defined for \(x\geq0\) by

\(f(x)=5-2\mathrm e^{-x}.\)

(a)(i) Find the domain of \(f^{-1}\).

(a)(ii) Solve

\(f^{-1}(x)=\sqrt{5x-4}.\)

(a)(iii) Sketch, on the same diagram, the graphs of \(y=f(x)\) and \(y=f^{-1}(x)\), showing the intercepts and asymptotes.

(b) The function \(g\) is defined for \(0\leq x\leq0.2\) by

\(g(x)=\frac{3}{1-x}.\)

Find and simplify an expression for \(f^{-1}g(x)\).