To determine what restriction should be applied to [tex]\( y = \tan x \)[/tex] for [tex]\( y = \arctan x \)[/tex] to be defined, let's analyze the behavior and properties of the tangent and arctangent functions:
1. Tangent Function [tex]\( \tan x \)[/tex]:
- The tangent function, [tex]\( \tan x \)[/tex], is periodic with a period of [tex]\( \pi \)[/tex].
- It has vertical asymptotes at [tex]\( x = \pm \frac{\pi}{2} \)[/tex], meaning that [tex]\( \tan x \)[/tex] is undefined at these points.
- Its range is [tex]\( (-\infty, \infty) \)[/tex].
2. Arctangent Function [tex]\( \arctan x \)[/tex]:
- The arctangent function, [tex]\( \arctan x \)[/tex], is the inverse of the tangent function but only over a specific interval where the tangent function is one-to-one and covers all possible real numbers.
- For [tex]\( \arctan x \)[/tex] to be defined, the original function [tex]\( \tan x \)[/tex] needs to cover every possible [tex]\( y \)[/tex]-value without any repetitions or ambiguities.
- The interval that satisfies this condition for [tex]\( \tan x \)[/tex] is [tex]\( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)[/tex]. However, for including all possible values (on the edges too), we consider [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex].
3. Why the Interval Matters:
- Within the interval [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex], the tangent function is continuous and strictly increasing.
- This makes the interval [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex] ideal for defining the arctangent function, as it ensures that [tex]\( y = \tan x \)[/tex] is one-to-one and can map back uniquely to [tex]\( x \)[/tex] through [tex]\( y = \arctan x \)[/tex].
Applying this understanding:
- To define [tex]\( y = \arctan x \)[/tex] properly, the range of [tex]\( y = \tan x \)[/tex] should be restricted.
From these explanations, the correct restriction is:
[tex]\[ \text{Restrict the range to } \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \][/tex]
Thus, the appropriate option is:
- Restrict the range to [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex]
