Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{e}^{{{(x-1)}^{2}}}}$,

Question

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined by $\mathrm{f}:x\mapsto {{e}^{{{(x-1)}^{2}}}}$, $x\in \mathbb{R}$, $\mathrm{g}:x\mapsto \frac{1}{2-x}$, $x\in \mathbb{R}$, $1\le xSketch the graph of $y=\mathrm{f}(x)$. [1] If the domain of $\mathrm{f}$ is restricted to $x\ge k$, state with a reason the least value of $k$ for which the function ${{\mathrm{f}}^{-1}}$ exists. [2] In the rest of the question, the domain of $\mathrm{f}$ is $x\ge k$, using the value of $k$ found in part (ii). Find ${{\mathrm{g}}^{-1}}(x)$ and show that the composite function ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$ exists. [4] Find the range of ${{\mathrm{g}}^{-1}}{{\mathrm{f}}^{-1}}$. [1]

Accepted Answer

(i) Minimum $(1,1)$, $y$-intercept $(0,\mathrm e)$ (ii) $k=1$ (iii) $\mathrm g^{-1}(x)=2-\dfrac1x$, $x\geq1$, and $\mathrm g^{-1}\mathrm f^{-1}$ exists (iv) $R_{\mathrm g^{-1}\mathrm f^{-1}}=[1,2)$

2021 RI P2 Q1

2021 RI P2 Q1

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