Q36 / 110 · VITEEE 2023 · Mathematics

Let \(\mathrm{f}:\left[-\frac{\pi}{3}, \frac{2 \pi}{3}\right] \rightarrow[0,4]\) be a function defined as \(f(x)=\sqrt{3} \sin x-\cos x+2\). Then, \(f^{-1}(x)\) is given by

  • A
    \(\sin ^{-1}\left(\frac{x-2}{2}\right)-\frac{\pi}{6}\)
  • Correct answer
    B
    \(\sin ^{-1}\left(\frac{x-2}{2}\right)+\frac{\pi}{6}\)
  • C
    \(\frac{2 \pi}{3}+\cos ^{-1}\left(\frac{\mathrm{x}-2}{2}\right)\)
  • D
    None of these

Solution

\(\begin{aligned} & \mathrm{f}:\left[-\frac{\pi}{3}, \frac{2 \pi}{3}\right] \rightarrow[0,4], \\ & \mathrm{f}(\mathrm{x})=\sqrt{3} \sin \mathrm{x}-\cos \mathrm{x}+2 \\ & \mathrm{f}(\mathrm{x})=2 \sin \left(\mathrm{x}-\frac{\pi}{6}\right)+2 \\ & \Rightarrow\left(\mathrm{f}^{-1}(\mathrm{x})\right)=\sin ^{-1}\left(\frac{\mathrm{x}-2}{2}\right)+\frac{\pi}{6} \end{aligned}\)

Hence, the correct answer is option: 'B'.

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