The function $\mathrm{f}$ is defined by \[\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\ri

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f} : x \rightarrow \left|\frac{2x + 4}{3 - x}\right|, \quad x \in \mathbb{R}, \quad x \neq 3.$$ Sketch the graph of $y = \mathrm{f}(x)$, giving the equations of any asymptotes and the coordinates of the points where the curve meets the axes.[3] Hence state the range of $\mathrm{f}$.[1] Explain why the function $\mathrm{f}^2$ does not exist.[1] The domain of $\mathrm{f}$ is further restricted to $x \leqslant a$, where $a$ is a constant. State the greatest value of $a$ such that the function $\mathrm{f}^{-1}$ exists.[1] Hence find $\mathrm{f}^{-1}(x)$ and state its domain.[4]

Accepted Answer

(a) asymptotes $x=3$, $y=2$, intercepts $(-2,0)$, $\left(0,\dfrac43\right)$ (b) $[0,\infty)$ (c) $3$ is in the range but not the domain, so $f^2$ does not exist (d) $a=-2$ (e) $f^{-1}(x)=\dfrac{3x+4}{x-2}$, domain $[0,2)$

N2023 P1 Q7

N2023 P1 Q7

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