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

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto -\ln (x-2)$, $x\in \mathbb{R}$, $2 < x\le 3$. Find ${{\mathrm{f}}^{-1}}(x)$ and state the domain and range of ${{\mathrm{f}}^{-1}}$.[4] Sketch on the same diagram the graphs of $y=\mathrm{f}(x)$, $y={{\mathrm{f}}^{-1}}(x)$ and $y={{\mathrm{f}}^{-1}}\mathrm{f}(x)$, giving the equations of any asymptotes and the exact coordinates of any points where the curves cross the $x$- and $y$- axes.[6] Explain why the x-coordinate of the point of intersection of the graph of $y=\mathrm{f}(x)$ and $y={{\mathrm{f}}^{-1}}(x)$ satisfies the equation $x+\ln (x-2)=0$, and find the value of this $x$-coordinate.[2]

Accepted Answer

(i) ${{\text{R}}_{{{\text{f}}^{-1}}}}={{\text{D}}_{\text{f}}}=(2,3]$ (ii) Asymptotes: $x=2$,$y=2$; Coordinates: $(3,3)$, $(0,3)$, $(3,0)$, $(2,2)$ (iii) Using GC, $x=2.12$ (3 s.f)

2020 TMJC Promo Q7

2020 TMJC Promo Q7

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