The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-4$ for \(x\in \ma

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto {{x}^{2}}+4x-4$ for $x\in \mathbb{R}$, $x<-2$. Explain why ${{\mathrm{f}}^{-1}}$ exists.[1] Sketch on the same diagram, the graphs of $y=\mathrm{f}\left( x \right)$, $y={{\mathrm{f}}^{-1}}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\,\mathrm{f}\left( x \right)$, stating the coordinates of endpoints and equations of asymptotes, if any.[3] Describe the relationship between the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.[1] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \ln \left( x+10 \right)$, where $x>-10$. Show that the composite function $\mathrm{gf}$ exists. Find the range of $\mathrm{gf}$.[3] Solve $\mathrm{f}\left( x \right)={{\mathrm{g}}^{-1}}\left( \ln \left( 6 \right) \right)$.[2]

Accepted Answer

(a)(i) $\mathrm f$ is strictly decreasing on $x-8$, and $\mathrm f^{-1}\mathrm f(x)=x$, $x<-2$, with open endpoints $(-2,-8)$, $(-8,-2)$, and $(-2,-2)$, intercepts $(-2-2\sqrt2,0)$ and $(0,-2-2\sqrt2)$, intersection $(-4,-4)$, no asymptotes (a)(iii) The graphs of $\mathrm f$ and $\mathrm f^{-1}$ are reflections of each other in $y=x$ (b)(i) $\mathrm{gf}$ exists because $R_{\mathrm f}=(-8,\infty)\subset D_{\mathrm g}=(-10,\infty)$, and $R_{\mathrm{gf}}=(\ln2,\infty)$ (b)(ii) $x=-4$

2022 SAJC Promo Q6

2022 SAJC Promo Q6

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