Question

Range of f(x) = $sin^{-1}x + cos^{-1}x + tan^{-1}x$ is

• A. [0, π]
• B. [$\frac{\pi}{2}$, π]
• C. [$\frac{\pi}{4}$, $\frac{3\pi}{4}$]
• D. [-$\frac{\pi}{2}$, $\frac{\pi}{2}$]

Answer 1

Answer: (C)

[$\frac{\pi}{4}$, $\frac{3\pi}{4}$]

Answer 2

The domain of $f(x) = \sin^{-1}x + \cos^{-1}x + \tan^{-1}x$ is $[-1, 1]$.

Using the identity $\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}$ for $x \in [-1, 1]$, the function simplifies to $f(x) = \frac{\pi}{2} + \tan^{-1}x$.

For $x \in [-1, 1]$, the range of $\tan^{-1}x$ is $[\tan^{-1}(-1), \tan^{-1}(1)] = [-\frac{\pi}{4}, \frac{\pi}{4}]$.

Therefore, the range of $f(x)$ is $[\frac{\pi}{2} - \frac{\pi}{4}, \frac{\pi}{2} + \frac{\pi}{4}] = [\frac{\pi}{4}, \frac{3\pi}{4}]$.