2023 ACJC Promo Q5
2023 ACJC Promo Q5
10 marks
Promo
- Functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by\({\mathrm{f}(x) = e^{x+3} - a, \; x \in \mathbb{R}, \; x \leq 4,}\)
\({\mathrm{g}(x) = 2x + 5 \text{ for}\hspace{0.5em} x \in \mathbb{R}.}\)where \(a\) is a constant.- Find \({{\mathrm{f}}^{-1}}\left( x \right)\) and state its domain, leaving your answer in terms of \(a\).[3]
- It is given that the composite function \({{\mathrm{g}}^{-1}}\mathrm{f}\) exists and \({{\mathrm{g}}^{-1}}\mathrm{f}\left( -3 \right)=15\). Without finding \({{\mathrm{g}}^{-1}}\mathrm{f}\), find the value of \(a\).[2]
- A curve has equation \(y=\mathrm{h}\left( x \right)\), where\(\mathrm{h}\left( x \right)\left\{ \begin{matrix} {{2}^{x}}\,\,\,\mathrm{for}\,\,\,0<x\le 2, \\ 6-\frac{1}{2}{{x}^{2}}\,\,\,\mathrm{for}\,\,\,2<x\le 3, \\ \end{matrix} \right.\)and that \(\mathrm{h}\left( x \right)=\mathrm{h}\left( x-3 \right)\) for all real values of \(x\).
- Find \(\mathrm{h}\left( 9 \right)\).[1]
- Sketch the graph of \(y=\mathrm{h}\left( x \right)\) for \(-1\le x\le 4\).[4]
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Answer:(a)(i) \(\mathrm f^{-1}(x)=\ln(x+a)-3\), domain \((-a,e^7-a]\). (a)(ii) \(a=-34\). (b)(i) \(\mathrm h(9)=\frac32\). (b)(ii) Sketch shown, including the open points \((0,1)\), \((3,1)\) and closed periodic endpoints.
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