Question

The domain of $\tan^{-1}x$ is

• A. $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$
• B. $(0,\pi)$
• C. $[-1,1]$
• D. $(-\infty,\infty)$

Hint:

Domain of $\tan^{-1}x$ is all real numbers while its range is $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

Answer 1

The Correct Option is D

Step 1: Recall the definition of inverse tangent function.
The inverse tangent function is written as \[ y=\tan^{-1}x \] This means \[ \tan y = x \] where \(y\) lies in the principal interval \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \]
Step 2: Understand the domain.
The domain of a function refers to all possible values of \(x\) for which the function is defined. For the tangent inverse function, the value of \(x\) can be any real number.
This means \[ x \in (-\infty,\infty) \]
Step 3: Range of the function.
Although the domain is all real numbers, the range is restricted to \[ \left(-\frac{\pi}{2},\frac{\pi}{2}\right) \]
Step 4: Conclusion.
Thus the inverse tangent function accepts every real value as input.
Final Answer: $\boxed{(-\infty,\infty)}$