If \(x \ge -1\), then the set of all values of \(\tan^{-1} x\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\left[-\displaystyle\frac{\pi}{4}, \displaystyle\frac{\pi}{2}\right)\)
The graph of \(y = \tan^{-1} x\) rises steadily from left to right, so \(\tan^{-1}\) is an increasing function. Its smallest value on \(x \ge -1\) is at \(x = -1\):
\( \displaystyle \tan^{-1}(-1) = -\frac{\pi}{4} \)
This value is reached, so \(-\frac{\pi}{4}\) is included.
As x grows without limit, \(\tan^{-1} x\) comes closer and closer to \(\frac{\pi}{2}\) but never reaches it, because \(\tan\frac{\pi}{2}\) is not defined. So \(\frac{\pi}{2}\) is excluded.
Hence the set of values is \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)\).
Why the other options are incorrect
Option (1) \(\left[-\frac{\pi}{4}, \frac{\pi}{2}\right]\) — Incorrect. \(\tan^{-1} x = \frac{\pi}{2}\) would need \(x = \tan\frac{\pi}{2}\), which does not exist.
Option (3) \(\left(-\frac{\pi}{4}, \frac{\pi}{2}\right)\) — Incorrect. \(x = -1\) is allowed, and it gives exactly \(-\frac{\pi}{4}\).
Option (4) \(\left[-\frac{\pi}{4}, \infty\right)\) — Incorrect. Every value of \(\tan^{-1}\) is less than \(\frac{\pi}{2}\).