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Mark the tick against the correct answer in the following: Range of \tan ^{-1} x is
A. \left(0, \frac{\pi}{2}\right)
B. \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)
C. \left[\frac{\pi}{2}, \frac{\pi}{2}\right]
D. None of these
CBSE | Class 12 | Inverse Trigonometric Functions | Maths | RS Aggarwal
Mark the tick against the correct answer in the following: Range of \tan ^{-1} x is
A. \left(0, \frac{\pi}{2}\right)
B. \left(\frac{-\pi}{2}, \frac{\pi}{2}\right)
C. \left[\frac{\pi}{2}, \frac{\pi}{2}\right]
D. None of these

Solution:

Option(B) is correct.
To Find: The range of \tan ^{-1} x
Here, the inverse function is given by y=f^{-1}(x)
The graph of the function y=\tan ^{-1}(x) can be obtained from the graph of Y=\tan x by interchanging x and y axes. i . e, if (a, b) is a point on Y=\tan x then (b, a) is the point on the function y=\tan ^{-1}(x)
Below is the Graph of the range of \tan ^{-1}(x)


From the graph, it is clear that the range of \tan ^{-1}(x) is restricted to any of the intervals like \left[-\frac{3 \pi}{2},-\frac{\pi}{2}\right],[ \left.-\frac{\pi}{2}, \frac{\pi}{2}\right],\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right] and so on. Hence the range is given by
\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)

i

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