The function $\mathrm{f}$ is defined by $\mathrm{f}:x\rightarrow x\left|x-3\right|,\ x\in\mathbb{R},\ x<m$. Show that $\mathrm{f}^{-1}$ does not exist if $m=4$.[1] State the largest possible value of $m$ for which $\mathrm{f}^{-1}$ exists.[1] Use the value of $m$ obtained in part (ii) for the following questions. Find $\mathrm{f}^{-1}$, stating its domain.[4] The composite function $\mathrm{fg}$ is defined by $\mathrm{fg}:x\rightarrow \frac{9}{4}-x^2,\ x\in\mathbb{R},\ x<0$. Find $\mathrm{g}(x)$.[2] The function $\mathrm{h}$ is defined by $\mathrm{h}:x\rightarrow \frac{3}{x-a},\ x\in\mathbb{R},\ x<a$. The graph of $y=\mathrm{k}(x)$ is transformed by a stretch with scale factor 2 parallel to the $x$-axis, followed by a translation of 10 units in the negative $y$-direction. The equation of the resulting curve is $y=\mathrm{h}(x)$. Find $\mathrm{k}(x)$.[2]

2014 HCI P2 Q4 (part)

10 marks

The function $\mathrm{f}$ is defined by $\mathrm{f}:x\rightarrow x\left|x-3\right|,\ x\in\mathbb{R},\ x<m$.

Use the value of $m$ obtained in part (ii) for the following questions.

Find $\mathrm{f}^{-1}$, stating its domain.[4]
The composite function $\mathrm{fg}$ is defined by $\mathrm{fg}:x\rightarrow \frac{9}{4}-x^2,\ x\in\mathbb{R},\ x<0$.
Find $\mathrm{g}(x)$.[2]
The function $\mathrm{h}$ is defined by $\mathrm{h}:x\rightarrow \frac{3}{x-a},\ x\in\mathbb{R},\ x<a$.
1. The graph of $y=\mathrm{k}(x)$ is transformed by a stretch with scale factor 2 parallel to the $x$-axis, followed by a translation of 10 units in the negative $y$-direction. The equation of the resulting curve is $y=\mathrm{h}(x)$.
  Find $\mathrm{k}(x)$.[2]

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