The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{1}{x-2}+5$, $x2$. Explain

Question

The function $\mathrm{f}$ is given by $\mathrm{f}:x\mapsto \frac{1}{x-2}+5$, $x>2$. Explain why ${{\mathrm{f}}^{-1}}$ exists.[1] Find ${{\mathrm{f}}^{-1}}$ in similar form.[3] Find $\mathrm{ff}$ in a similar form and the exact range of $\mathrm{ff}$.[3]

Accepted Answer

(i) $\mathrm f^{-1}$ exists because $\mathrm f$ is one-to-one (ii) $\mathrm f^{-1}:x\mapsto\dfrac1{x-5}+2$, $x>5$ (iii) $\mathrm{ff}:x\mapsto\dfrac{x-2}{3x-5}+5$, $x>2$, with $R_{\mathrm{ff}}=\left(5,\dfrac{16}{3}\right)$

2023 RVHS Promo Q3

2023 RVHS Promo Q3

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