The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{2x-1}{2x-2},\quad x

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \frac{2x-1}{2x-2},\quad x < 1$. Define ${{\mathrm{f}}^{-1}}$ in a similar form. Hence or otherwise find ${{\mathrm{f}}^{2}}(x)$ and state the value of ${{\mathrm{f}}^{1965}}(0.5)$.[5] The function $\mathrm{g}$ is defined by $\mathrm{g}:x\mapsto \sqrt{4+2x},\quad x\in \mathbb{R},\ \ 0\le x < 16.$ Sketch the graph of $y=\mathrm{g}(x)$. Your sketch should indicate the coordinates of the end-points.[2] On the same diagram of part (i), sketch the graph of $y={{\mathrm{g}}^{-1}}(x)$, showing the relationship between $\mathrm{g}$ and ${{\mathrm{g}}^{-1}}$.[2] Find the values of $x$ satisfying the equation ${{\mathrm{g}}^{-1}}\mathrm{g}(x)=\mathrm{g}{{\mathrm{g}}^{-1}}(x)$.[1]

Accepted Answer

(a) $\mathrm f^{-1}(x)=\dfrac{2x-1}{2x-2}$, $x<1$; $\mathrm f^2(x)=x$, and $\mathrm f^{1965}(0.5)=0$ (b)(i) Closed endpoint $(0,2)$, open endpoint $(16,6)$ (b)(ii) $\mathrm g^{-1}(x)=\dfrac{x^2-4}{2}$, $2\leq x<6$, reflected in $y=x$ (b)(iii) $2\leq x<6$

2015 NYJC P2 Q2

2015 NYJC P2 Q2

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