A piecewise function is given as \({\mathrm{f}(x) = \begin{cases} 3 - x & \mathrm{for } 1 \leq x \le

Question

A piecewise function is given as ${\mathrm{f}(x) = \begin{cases} 3 - x & \mathrm{for } 1 \leq x \leq 3, \ \sqrt{4 - (x - 5)^2} & \mathrm{for } 3 < x \leq 5 \end{cases}}$ And that $\mathrm{f}\left( x \right)=\mathrm{f}\left( x-4 \right)$ for all real values of $x$. Find the exact values of $\mathrm{f}\left( 8 \right)$ and $\mathrm{f}\left( -10 \right)$.[2] Another function $\mathrm{g}$ is defined as $\mathrm{g}\left( x \right)=\ln x$, ${{e}^{2}} < x < {{e}^{4}}$. Show that $\mathrm{fg}$ exists and find the exact range of $\mathrm{fg}$.[3]

Accepted Answer

(i) $\mathrm{f}\left( 8 \right)=\sqrt{3}$; $\mathrm{f}\left( -10 \right)=1$ (ii) ${{\mathrm{R}}_{\mathrm{fg}}}=\left[ 0,\sqrt{3} \right)$

2022 HCI J1 BT Q1

2022 HCI J1 BT Q1

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