Write the domain of `f(x) = tan^(-1)x`

Answer

To find the domain of the function \( f(x) = \tan^{-1}(x) \), we need to determine the set of all possible input values (x-values) for which the function is defined.

### Step-by-Step Solution:

1. **Understanding the Function**: The function \( f(x) = \tan^{-1}(x) \) is the inverse of the tangent function. The tangent function, \( \tan(\theta) \), is defined for all real numbers \( \theta \) except for odd multiples of \( \frac{\pi}{2} \).

2. **Range of the Tangent Function**: The tangent function can take any real number as its output. Therefore, the range of \( \tan(\theta) \) is all real numbers, which means that for every real number \( y \), there exists a \( \theta \) such that \( y = \tan(\theta) \).

3. **Domain of the Inverse Function**: Since the range of the tangent function is all real numbers, the domain of its inverse function \( \tan^{-1}(x) \) must also be all real numbers. This means that you can input any real number into \( \tan^{-1}(x) \).

4. **Conclusion**: Therefore, the domain of \( f(x) = \tan^{-1}(x) \) is:
\[ \text{Domain of } f(x) = (-\infty, \infty) \]

### Final Answer:
The domain of \( f(x) = \tan^{-1}(x) \) is \( (-\infty, \infty) \).