The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \ln {{(x+1)}^{2}}+2$, $x\in \mathbb{R}$

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \ln {{(x+1)}^{2}}+2$, $x\in \mathbb{R}$, $x\ge 0$. Show that $\mathrm{f}$ has an inverse. [2] Find ${{\mathrm{f}}^{-1}}(x)$ and state the domain of ${{\mathrm{f}}^{-1}}$. [3] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \frac{1-x}{x+1}$, $x\in \mathbb{R}$, $x
e -1$. Given that ${{\mathrm{g}}^{-1}}$ exists, show that ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$ exists, and find the range of ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$. [3] Find the exact value of $x$ such that ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}(x)=\frac{1}{2}$. [3]

Accepted Answer

(a) f⁻¹ exists; (b) f⁻¹(x) = 1 + e^((x-2)/2), Df⁻¹ = [2, ∞); (c) g⁻¹f⁻¹ exists, Rg⁻¹f⁻¹ = (-1, 1]; (d) x = ln(16/9) + 2

2023 HCI P1 Q9

2023 HCI P1 Q9

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