The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{{{x}^{2}}}{x-4}$, $x\in \mathbb{R

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{{{x}^{2}}}{x-4}$, $x\in \mathbb{R}$, $4 Find ${{\mathrm{f}}^{-1}}\left( x \right)$ and write down the domain of ${{\mathrm{f}}^{-1}}$. On the same axes, sketch the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$. [4] The region $R$ is bounded by the curve $y={{\mathrm{f}}^{-1}}\left( x \right)$, the lines $y=5$, $y=8$ and the $y$-axis. Find the exact area of $R$. [3]

Accepted Answer

(a) $\displaystyle \mathrm{f}^{-1}(x)=\frac{x-\sqrt{x^2-16x}}{2}$, domain $[16,\infty)$ (b) $\displaystyle\frac{63}{2}+32\ln2\text{ units}^2$

2024 DHS P1 Q5

2024 DHS P1 Q5

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