The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{1+x}-2$, fo

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto \frac{1}{1+x}-2$, for $x\in \mathbb{R}$, $x\ge 0$, $\mathrm{g}:x\mapsto {{\left( x+\frac{4}{3} \right)}^{2}}-\frac{1}{5}$ for $x\in \mathbb{R}$. Determine whether the composite function $\mathrm{fg}$ exists. [2] Sketch the graph of $y=\mathrm{f}\left( x \right)$ and find the range of $\mathrm{f}$. [2] Find the exact range of $\mathrm{gf}$. [2] The function $\mathrm{h}$ is defined by $\mathrm{h}:x\mapsto {{\left( x+1 \right)}^{2}}\left( x-2 \right)$, $x\in \mathbb{R}$. Explain why the function ${{\mathrm{h}}^{-1}}$ does not exist. [1] For the rest of the question, the domain of $\mathrm{h}$ is restricted to $x\in \left[ -1,1 \right]$. State the domain of ${{\mathrm{h}}^{-1}}$. [1] Sketch on the same diagram the graphs of $y=\mathrm{h}\left( x \right)$, $y={{\mathrm{h}}^{-1}}\left( x \right)$ and $y={{\mathrm{h}}^{-1}}\mathrm{h}\left( x \right)$. [3]

Accepted Answer

R_{gf} = \left[-\frac{1}{5}, \frac{11}{45}\right)

2020 NJC Promo Q9

2020 NJC Promo Q9

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