The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 4{{\left( x-1 \right)}^{3}}+1,$ \(x

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto 4{{\left( x-1 \right)}^{3}}+1,$ $x\in \mathbb{R}$, $0 < x < 2$ . Define ${{\mathrm{f}}^{-1}}$ in a similar form.[3] Sketch on the same diagram the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$, giving the coordinates of the end points clearly. State the relationship between the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$.[4] Solve $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$.[1] State the solution set for $\mathrm{f}{{\mathrm{f}}^{-1}}\left( x \right)={{\mathrm{f}}^{-1}}\mathrm{f}\left( x \right)$.[1]

Accepted Answer

(i) ${{\mathrm{f}}^{-1}}:x\mapsto \sqrt[3]{\frac{x-1}{4}}+1,x\in \mathbb{R},-3<x<5$ (ii) Reflection in the line $y=x$. (iii) $x=1,\frac{1}{2},\frac{3}{2}$ (iv) $\left( 0,2 \right)$

2016 NYJC Promo Q7

2016 NYJC Promo Q7

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