The function $\mathrm{f}$ is defined by\[\mathrm{f} : x \mapsto \frac{ax - 3}{5x - 4}, \text{ for}

Question

The function $\mathrm{f}$ is defined by$$\mathrm{f} : x \mapsto \frac{ax - 3}{5x - 4}, \text{ for}\hspace{0.5em} x \in \mathbb{R}, x \neq \frac{4}{5}, \text{ and}\hspace{0.5em} a \in \mathbb{R}$$ Find the range of $\mathrm{f}$ in terms of $a$.[1] The function $\mathrm{f}$ is such that $\mathrm{f}(x) = \mathrm{f}^{-1}(x)$ for all $x$ in the domain of $\mathrm{f}$. Find the value of $a$.[2] Hence find $\mathrm{f}^{2025}(2)$.[2] Another function $\mathrm{g}$ is defined by$$\mathrm{g} : x \mapsto \left(x - \frac{1}{5}\right)^2 + \frac{2}{5}, \text{ for}\hspace{0.5em} x \in \mathbb{R}, 0 < x < \frac{3}{5}$$ Find the range of $\mathrm{fg}$.[2]

Accepted Answer

(a) If $a\ne\dfrac{15}{4}$, $R_{\mathrm f}=\mathbb R\setminus\left\{\dfrac a5\right\}$; if $a=\dfrac{15}{4}$, $R_{\mathrm f}=\left\{\dfrac34\right\}$ (b) $a=4$ (c) $\dfrac56$ (d) $R_{\mathrm{fg}}=\left(\dfrac{19}{30},\dfrac7{10}\right]$

2025 HCI P1 Q8

2025 HCI P1 Q8

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