To determine the restriction that should be applied to [tex]\( y = \tan x \)[/tex] for [tex]\( y = \arctan x \)[/tex] to be properly defined, we need to understand the relationship between the tangent function and its inverse, the arctangent function.
The tangent function, [tex]\( \tan x \)[/tex], is periodic with a period of [tex]\( \pi \)[/tex] and has vertical asymptotes at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] where [tex]\( k \)[/tex] is an integer. Importantly, [tex]\( \tan x \)[/tex] is not one-to-one over its entire domain, which means it does not pass the horizontal line test over a full period of [tex]\( \pi \)[/tex].
For an inverse function to be properly defined, its corresponding original function must be one-to-one over the restricted interval. This ensures that every [tex]\( y \)[/tex] value in the range of [tex]\( \arctan x \)[/tex] maps to exactly one [tex]\( x \)[/tex] value in the domain of [tex]\( \tan x \)[/tex].
The most common and useful interval to restrict [tex]\( \tan x \)[/tex] is [tex]\( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \)[/tex] or [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex] because within this interval, [tex]\( \tan x \)[/tex] is one-to-one and continuous.
Among the given options, the appropriate restriction is:
- Restrict the range to [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex]
This option ensures that [tex]\( y = \tan x \)[/tex] is restricted in such a way that [tex]\( y = \arctan x \)[/tex] can be properly defined over this interval, covering all possible [tex]\( y \)[/tex] values from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
Therefore, the correct restriction is:
[tex]\[ \boxed{2} \][/tex] which corresponds to restricting the range to [tex]\( \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right] \)[/tex].
