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

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \lambda x-{{x}^{2}},\quad x\in \mathbb{R},\quad x>\frac{\lambda }{2},$ where $\lambda >2$. Find, in terms of $\lambda $, ${{\mathrm{f}}^{-1}}\left( x \right)$, stating the domain of ${{\mathrm{f}}^{-1}}$. [4] On the same diagram, sketch the graphs of $y=\mathrm{f}\left( x \right)$ and $y={{\mathrm{f}}^{-1}}\left( x \right)$, showing their graphical relationship clearly. [2] Find, in terms of $\lambda $, the solution of the equation $\mathrm{f}\left( x \right)={{\mathrm{f}}^{-1}}\left( x \right)$. [3]

Accepted Answer

$x=\lambda-1$ for all $\lambda>2$. When $3<\lambda<1+\sqrt5$, also $x=\dfrac{\lambda+1\pm\sqrt{(\lambda-3)(\lambda+1)}}2$.

2015 PJC P2 Q1

2015 PJC P2 Q1

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