Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by \(\mathrm{f}:x\mapsto {{x}^{3}}-7{{x}^{2}

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{x}^{3}}-7{{x}^{2}}-5x+11$, $x\in \mathbb{R}$, $x\ge k$, $\mathrm{g}:x\mapsto {{\left( x+1 \right)}^{2}}+2$,$x\in \mathbb{R}$. Let $k=1$. Show that $\mathrm{f}$ does not have an inverse.[2] Determine whether the composite function $\mathrm{fg}$ exists.[2] Find the value of $k$ given that ${{\mathrm{f}}^{-1}}$ exists and that the domain of ${{\mathrm{f}}^{-1}}$ is $x\in \mathbb{R}$, $x\ge -24$.[2] Let $k=6$. Show algebraically that $\mathrm{f}'(x)>0$ for all values of $x$ in the domain of $\mathrm{f}$.[2] Solve the equation $\mathrm{g}{{\mathrm{f}}^{-1}}(x)=83$.[3]

Accepted Answer

(i)(a) The line $y=0$ cuts the graph of $y=\mathrm{f}(x)$ at $x=1$ and $x=7.47$, so $\mathrm{f}$ is not one-to-one, ${{\mathrm{f}}^{-1}}$ does not exist. (i)(b) Since ${{D}_{\mathrm{f}}}=[1,\infty ),\,{{R}_{\mathrm{g}}}\subseteq {{D}_{\mathrm{f}}}$. So $\mathrm{fg}$ exists. (ii) $k=7$ (iii)(b) $x=\mathrm{f}(8)=35$

2020 EJC Promo Q10

2020 EJC Promo Q10

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