Functions $\mathrm{g}$ and $\mathrm{h}$ are defined as $\mathrm{g}:x\mapsto 7{{\mathrm{e}}^{-x}}-\ma

Question

Functions $\mathrm{g}$ and $\mathrm{h}$ are defined as $\mathrm{g}:x\mapsto 7{{\mathrm{e}}^{-x}}-\mathrm{e}$, $x\in \mathbb{R}$, $x>0$, $\mathrm{h}:x\mapsto {{x}^{2}}+2x+5$, $x\in \mathbb{R}$, $x\le -1$. Determine if the composite function $\mathrm{hg}$ exists. [2] Write down an expression for $\mathrm{gh}\left( x \right)$ and evaluate $\mathrm{gh}\left( -1 \right)$. [2] Explain clearly why $\mathrm{gh}\left( 0 \right)$ is not defined. [1] Determine the exact range of $\mathrm{gh}$. [2]

Accepted Answer

(i) $	herefore {{\mathrm{R}}_{\mathrm{g}}} ot\subset {{\mathrm{D}}_{\mathrm{h}}}$, $\mathrm{hg}$ doesn’t exist. (ii) ${\mathrm{gh}\left( x ight)=7{{\mathrm{e}}^{-\left( {{x}^{2}}+2x+5 ight)}}-\mathrm{e}}$ ; $\mathrm{gh}\left( -1 ight)=7{{\mathrm{e}}^{-4}}-\mathrm{e}$ 	ext{ or } $-2.59$ ($3$ s.f.) (iii) Since $0 otin {{\mathrm{D}}_{\mathrm{gh}}}={{\mathrm{D}}_{\mathrm{h}}}=\left( -\propto ,-1 ight]$, therefore $\mathrm{gh}\left( 0 ight)$ is not defined. (iv) ${{\mathrm{R}}_{\mathrm{gh}}}=\left( -\mathrm{e,7}{{\mathrm{e}}^{-4}}-\mathrm{e} ight]$

2022 EJC J1 MYE Q3

2022 EJC J1 MYE Q3

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