2021 RI P2 Q1

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Timothy Gan

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2021 RI P2 Q1

Functions $\text{f}$ and $\text{g}$ are defined by

$\text{f}:x\mapsto {{e}^{{{(x-1)}^{2}}}}$, $x\in \mathbb{R}$,
$\text{g}:x\mapsto \frac{1}{2-x}$, $x\in \mathbb{R}$, $1\le x<2$.

(i)

Sketch the graph of $y=\text{f}(x)$.

[1]

(ii)

If the domain of $\text{f}$ is restricted to $x\ge k$, state with a reason the least value of $k$ for which the function ${{\text{f}}^{-1}}$ exists.

[2]

In the rest of the question, the domain of $\text{f}$ is $x\ge k$, using the value of $k$ found in part (ii).

(iii)

Find ${{\text{g}}^{-1}}(x)$ and show that the composite function ${{\text{g}}^{-1}}{{\text{f}}^{-1}}$ exists.

[4]

(iv)

Find the range of ${{\text{g}}^{-1}}{{\text{f}}^{-1}}$.

[1]

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Published: 13th December 2022

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