A function $\mathrm{h}$ is self-inverse if \(\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\l

Question

A function $\mathrm{h}$ is self-inverse if $\mathrm{h}\left( x \right)={{\mathrm{h}}^{-1}}\left( x \right)$ for all $x$ in the domain of $\mathrm{h}$. The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}\left( x \right)=2+\frac{1}{x-2}$ $\,\,\,\,\,\,\,\,\,$ for $\,\,\,\,\,\,\,\,\,$ $x\in \mathbb{R},x>2$ $\mathrm{g}\left( x \right)=\left\{ \begin{matrix}3x-1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\le 1 \ {{\left( x-1 \right)}^{2}}+2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,\,\,\,\,\,\,\,\,\,\,\,\,x>1 \\end{matrix} \right.$ Sketch the graph of $y=\mathrm{f}(x)$. With the aid of your graph, explain why $\mathrm{f}$ has an inverse.[2] Show that $\mathrm{f}$ is self-inverse and find ${{\mathrm{f}}^{2}}\left( x \right)$.[4] Hence, or otherwise, evaluate ${{\mathrm{f}}^{2025}}\left( 4 \right)$.[2] Find an expression for $\mathrm{gf}$ and state its domain.[3] Find the range of $\mathrm{gf}$.[2]

Accepted Answer

(a) The upper-right branch with asymptotes $x=2$ and $y=2$; every horizontal line meets it at most once, so $\mathrm f^{-1}$ exists (b) $\mathrm f^{-1}=\mathrm f$, and $\mathrm f^2(x)=x$ for $x>2$ (c) $\dfrac52$ (d) $\mathrm{gf}(x)=3+\dfrac2{x-2}+\dfrac1{(x-2)^2}$, with $D_{\mathrm{gf}}=(2,\infty)$ (e) $R_{\mathrm{gf}}=(3,\infty)$

2024 CJC Promo Q11

2024 CJC Promo Q11

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