Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\to {{x}^{2}}$, \( \qquad

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\to {{x}^{2}}$, $ \qquad x<0$. $\mathrm{g}:x\to \frac{1}{x}$, $ \qquad x>0$. Explain why the composite function $\mathrm{gf}$ exists.[1] Find the exact value of ${{\mathrm{f}}^{-1}}{{\mathrm{g}}^{-1}}\left( 3 \right)$. Show your workings clearly.[3] For real values $a$, the function $h$ is defined by $\mathrm{h}:x\mapsto ax-\frac{1}{x}$, $x<0$. If $a$ is negative, explain clearly with a well-labelled diagram, why ${{\mathrm{h}}^{-1}}$ does not exist.[4] If $a=1$, find ${{\mathrm{h}}^{-1}}$ in similar form.[3]

Accepted Answer

(a)(i) $\mathrm R_f=(0,\infty)=\mathrm D_g$, so $gf$ exists. (a)(ii) $-\frac{\sqrt3}{3}$ (b)(i) The graph has minimum $2\sqrt{-a}$ and is not one-to-one, so $h^{-1}$ does not exist. (b)(ii) $h^{-1}(x)=\frac{x-\sqrt{x^2+4}}2$, $x\in\mathbb R$

2022 ASRJC P1 Q8

2022 ASRJC P1 Q8

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