A logarithmic involution

(a) The curve $y=f(x)$ has horizontal asymptote $y=k$ and cuts the axes at $(a, 0)$ and $(0, b)$, where $a, b, k$ are non-zero constants. Given that $f^{-1}$ exists, state if possible the coordinates where the following curves cut the axes and their asymptotes: (i) $y=f(2x-3)$, (ii) $y=f^{-1}(x)$.2 marks

(b) $g:x\mapsto\ln\left(\frac{e^x+5}{e^x-1}\right)$ for $x\in\mathbb R$, $x>0$.

(i) Find $g^{-1}(x)$ and state its domain.3 marks

The function $h:x\mapsto1+\sqrt{9-(x-2)^2}$ for $x\in\mathbb R$, $-1\le x\le5$.

(ii) Find the exact solutions of $g^{2021}h(x)=\ln2$ in simplest form.3 marks

[2021/NYJC/Prelim/I/5]