The function $\mathrm{f}$, with domain the set of all real values, is given by \({\mathrm{f}(x)=\b

Question

The function $\mathrm{f}$, with domain the set of all real values, is given by ${\mathrm{f}(x)=\begin{cases} 4-x^2 & \text{for}\hspace{0.5em} 0 < x \leq 2, \ 2x-4 & \text{for}\hspace{0.5em} 2 < x \leq 4, \end{cases}}$ and that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x+4 \right)$. Find $\mathrm{f}\left( 45 \right)$.[1] Sketch the graph of $y=\mathrm{f}\left( x \right)$ for $-4\le x\le 7$ and state the range of $\mathrm{f}$.[3] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto {{\left( x-1 \right)}^{2}}+2$ for $x\in \mathbb{R}$, $0 < x < 1$. Explain why the composite function $\mathrm{gf}$ does not exist.[1] Find an expression for $\mathrm{fg}\left( x \right)$ and hence, or otherwise, find ${{\left( \mathrm{fg} \right)}^{-1}}\left( \frac{1}{2} \right)$.[4]

Accepted Answer

(i) $3$ (ii) $R_f=[0,4]$ (iii) $R_f\nsubseteq D_g$, so $gf$ does not exist. (iv) $fg(x)=2(x-1)^2$, $(fg)^{-1}\left(\frac12\right)=\frac12$

2023 JPJC P1 Q5

2023 JPJC P1 Q5

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