The function $\mathrm{f}$ is defined by \(\mathrm{f}:x\mapsto 4+\frac{1}{{{\left( x-1 \right)}^{2}

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 4+\frac{1}{{{\left( x-1 \right)}^{2}}+1}$, $x\ge 0$. Sketch the graph of $y=\mathrm{f}\left( x \right)$, indicating clearly the coordinates of any stationary points, the value of the $y-$intercept, and the equation of any asymptotes.[3] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \sin x$, $x\in \left[ 0,2\pi \right]$. State the exact coordinates of two points on the graph of $y=\mathrm{g}\left( x \right)$, which demonstrate that $\mathrm{g}$ does not have an inverse.[1] Explain clearly why the composite function $\mathrm{gf}$ exists.[2] Using a suitably-labelled sketch of the graph of $y=\mathrm{g}\left( x \right)$, find the range of $\mathrm{gf}$.[3]

Accepted Answer

(a) Stationary maximum $(1,5)$, $y$-intercept $\left(0,\frac92\right)$, asymptote $y=4$ (b) $(0,0)$ and $(\pi,0)$ (c) $\mathrm R_f=(4,5]\subseteq[0,2\pi]=\mathrm D_g$, so $gf$ exists. (d) $\mathrm R_{gf}=[-1,\sin4)$

2023 EJC P1 Q8

2023 EJC P1 Q8

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