Let's analyze the given problem step-by-step to evaluate the truthfulness of the statement.
1. Function Definition:
- The function [tex]\( f(x) = \tan^{-1}(x) \)[/tex] is the inverse of the tangent function. This is commonly denoted as the arctangent function.
2. Domain of [tex]\( f(x) \)[/tex]:
- The arctangent function [tex]\( \tan^{-1}(x) \)[/tex] takes any real number as its input. Hence, the domain of [tex]\( f(x) \)[/tex] is all real numbers, [tex]\((-\infty, \infty)\)[/tex].
3. Range of [tex]\( f(x) \)[/tex]:
- The arctangent function [tex]\( \tan^{-1}(x) \)[/tex] outputs values that lie within a specific interval.
- For typical usage, the range of [tex]\( \tan^{-1}(x) \)[/tex] is [tex]\( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)[/tex]. This is the standard range for the principal values of the arctangent function.
4. Given Range in the Statement:
- The statement claims that the range is [tex]\([0, \pi]\)[/tex]. However, this is incorrect.
- The correct standard range for the arctangent function is [tex]\(\left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex], and definitely not [tex]\([0, \pi]\)[/tex].
Based on this understanding, the statement that the range of [tex]\( f(x) = \tan^{-1}(x) \)[/tex] is [tex]\([0, \pi]\)[/tex] is false.
Thus, the correct answer is:
B. False
