Self-inverse functions alternate by parity

A function $f$ is said to be a self-inverse function if $f(x)=f^{-1}(x)$. The function $f$ is defined by $f(x)=\frac{ax+b}{cx-2}$, $x\in\mathbb R$, $x\ne2/c$, where $a, b, c, d$ are non-zero constants.

(i) By finding $f^{-1}(x)$, show that $a=2$ for $f$ to be self-inverse.2 marks

For the rest use $a=2, b=3, c=5$. (ii) Find $f^2(x)$. 1 mark (iii) Evaluate $f^{71}(4)$.2 marks

The function $g$ is defined by $g(x)=2x^2-3$, $x\in\mathbb R$. (iv) Explain why composite $fg$ does not exist. 2 marks (v) Find exact solutions of $gf(x)=5$.3 marks

[2020/CJC/Promo/10]