Two functions are defined on the domain $0\le x\le \pi $ by $\mathrm{f}:x\mapsto \sin x$ and $\mathr

Question

Two functions are defined on the domain $0\le x\le \pi $ by $\mathrm{f}:x\mapsto \sin x$ and $\mathrm{g}:x\mapsto \cos x$. Explain why one of these functions has an inverse while the other does not. [2] If the domain is restricted to $0\le x\le k$, the inverse exists for both functions. State the greatest possible value of $k$ in exact form. [1] Evaluate ${{\mathrm{f}}^{-1}}\mathrm{g}\left( \frac{\pi }{3} ight)$ in exact form. [1] The function $\mathrm{h}$ is defined by: $\mathrm{h}:x\mapsto \frac{x+a}{2x+b}$, for $x\in \mathbb{R}$, $x
e k$. Give an expression for $k$ and explain why this value of $x$ has to be excluded from the domain of $\mathrm{h}$. [2] The function $\mathrm{h}$ is such that $\mathrm{h}\left( x ight)={{\mathrm{h}}^{-1}}\left( x ight)$ for all $x$ in the domain of $\mathrm{h}$. Determine the possible values of $a$ and $b$. [3] Find an expression for ${{\mathrm{h}}^{-1}}\left( -4 ight)$. [1]

Accepted Answer

(a)(i) g⁻¹ exists, f⁻¹ does not; (a)(ii) k=π/2; (a)(iii) π/6; (b)(i) k=−b/2; (b)(ii) b=−1, a≠−1/2; (b)(iii) (4−a)/9

2022 NYJC CT Q7

2022 NYJC CT Q7

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