To determine which function has the range [tex]\(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)[/tex], let's analyze the range of each of the given inverse trigonometric functions.
1. [tex]\(y = \cos^{-1} x\)[/tex] (arccos(x))
- The arccosine function is the inverse function of the cosine function.
- It is defined for [tex]\(x\)[/tex] in the interval [tex]\([-1, 1]\)[/tex].
- The range of arccos(x) is [tex]\([0, \pi]\)[/tex].
2. [tex]\(y = \tan^{-1} x\)[/tex] (arctan(x))
- The arctangent function is the inverse function of the tangent function.
- It is defined for all real numbers [tex]\(x\)[/tex].
- The range of arctan(x) is [tex]\(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)[/tex].
3. [tex]\(y = \sec^{-1} x\)[/tex] (arcsec(x))
- The arcsec function is the inverse of the secant function.
- It is defined for [tex]\(x \leq -1\)[/tex] or [tex]\(x \geq 1\)[/tex], i.e., [tex]\(|x| \geq 1\)[/tex].
- The range of arcsec(x) is [tex]\([0, \pi]\)[/tex] except [tex]\(\pi/2\)[/tex] (i.e., [tex]\(\left[0, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \pi\right)\)[/tex]).
4. [tex]\(y = \csc^{-1} x\)[/tex] (arccsc(x))
- The arccsc function is the inverse of the cosecant function.
- It is defined for [tex]\(x \leq -1\)[/tex] or [tex]\(x \geq 1\)[/tex], i.e., [tex]\(|x| \geq 1\)[/tex].
- The range of arccsc(x) is [tex]\(\left[-\frac{\pi}{2}, 0\right) \cup \left(0, \frac{\pi}{2}\right]\)[/tex].
Given these ranges, the function [tex]\(y = \tan^{-1} x\)[/tex] (arctan(x)) is the only one that has the range [tex]\(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)[/tex].
Thus, the function with the range [tex]\(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)[/tex] is:
[tex]\[ y = \tan^{-1} x \][/tex]
Therefore, the correct answer is:
[tex]\[ 2 \][/tex]
