The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x-1$. Show

Question

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\mapsto \dfrac{4x-1}{x+1},\ x>-1$. Show that $\mathrm{f}$ has an inverse.[2] Find $\mathrm{f}^{-1}(x)$ and state its domain.[3] State the range of $\mathrm{f}^{-1}$.[1] State the equation of the line in which the graph of $y=\mathrm{f}(x)$ must be reflected in order to obtain the graph of $y=\mathrm{f}^{-1}(x)$.[1] Hence, or otherwise, find the exact solutions of the equation $\mathrm{f}(x)=\mathrm{f}^{-1}(x)$.[3]

Accepted Answer

$\mathrm{f}^{-1}(x)=\dfrac{x+1}{4-x}$, domain $(-\infty,4)$, range $(-1,\infty)$; reflection $y=x$; $x=\dfrac{3\pm\sqrt5}{2}$

2025 MI PU1 Promo Q9

2025 MI PU1 Promo Q9

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