Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
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]
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]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
