The function \(f\) is defined by

\(f(x)=\ln(2x+1)\qquad\text{for }x\geq0.\)

(a) Sketch the graph of \(y=f(x)\) and hence sketch the graph of \(y=f^{-1}(x)\).

The function \(g\) is defined by

\(g(x)=(x-4)^2+1\qquad\text{for }x\leq4.\)

(b)(i) Find an expression for \(g^{-1}(x)\) and state its domain and range.

(b)(ii) Find and simplify an expression for \(fg(x)\).

(b)(iii) Explain why the function \(gf\) does not exist.