The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows $\mathrm{f}:x\mapsto \sin x$, $x\

Question

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows $\mathrm{f}:x\mapsto \sin x$, $x\in \mathbb{R}$, $-\pi \le x\le 2\pi $. $\mathrm{g}:x\mapsto \cos x$, $x\in \mathbb{R}$, $0\le x\le 2\pi $. Sketch the graph of $y=\mathrm{f}\left( x \right)$ and explain why ${{\mathrm{f}}^{-1}}$ does not exist. The function $\mathrm{h}$ is a restriction on $\mathrm{f}$ such that $\mathrm{f}$ and $\mathrm{h}$ share the same range, and both the functions ${{\mathrm{h}}^{-1}}$ and $\mathrm{g}{{\mathrm{h}}^{-1}}$ exists. Write down the domain of $\mathrm{h}$. Determine the range of $\mathrm{g}{{\mathrm{h}}^{-1}}$. Find ${{\mathrm{h}}^{-1}}$ in a similar form.

Accepted Answer

(i) f⁻¹ does not exist; (ii) D_h = [π/2, 3π/2]; (iii) R_{gh⁻¹} = [-1, 0]; (iv) h⁻¹: x → π − sin⁻¹x

SAJC Functions Tutorial Q11

SAJC Functions Tutorial Q11

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