Discover the answers you need at Westonci.ca, a dynamic Q&A platform where knowledge is shared freely by a community of experts. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To determine which function has the range [tex]\((-\infty, -2] \cup [0, \infty)\)[/tex], we need to analyze the range of each function one by one.
Option A: [tex]\( y = \csc(x) - 1 \)[/tex]
First, recall that [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex], which makes the range of [tex]\( \csc(x) \)[/tex] be [tex]\((-\infty, -1] \cup [1, \infty)\)[/tex]. Now, if we subtract 1 from [tex]\( \csc(x) \)[/tex]:
- For [tex]\( \csc(x) \)[/tex] values in [tex]\((-\infty, -1]\)[/tex]: [tex]\( \csc(x) - 1 \leq -2 \)[/tex]
- For [tex]\( \csc(x) \)[/tex] values in [tex]\([1, \infty)\)[/tex]: [tex]\( \csc(x) - 1 \geq 0 \)[/tex]
Thus, the range of [tex]\( y = \csc(x) - 1 \)[/tex] is [tex]\((-\infty, -2] \cup [0, \infty)\)[/tex].
Option B: [tex]\( y = \sec(x) + 1 \)[/tex]
Next, note that [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex], which makes the range of [tex]\( \sec(x) \)[/tex] be [tex]\((-\infty, -1] \cup [1, \infty)\)[/tex]. Adding 1 to [tex]\( \sec(x) \)[/tex]:
- For [tex]\( \sec(x) \)[/tex] values in [tex]\((-\infty, -1]\)[/tex]: [tex]\( \sec(x) + 1 \leq 0 \)[/tex]
- For [tex]\( \sec(x) \)[/tex] values in [tex]\([1, \infty)\)[/tex]: [tex]\( \sec(x) + 1 \geq 2 \)[/tex]
So, the range of [tex]\( y = \sec(x) + 1 \)[/tex] is [tex]\((-\infty, 0] \cup [2, \infty)\)[/tex], which does not match the required range.
Option C: [tex]\( y = \cos(x + 1) \)[/tex]
For [tex]\( y = \cos(x + 1) \)[/tex], the function essentially has the same range as [tex]\( \cos(x) \)[/tex] since horizontal shifts do not affect the range of cosine. The range of [tex]\( \cos(x) \)[/tex] is [tex]\([-1, 1]\)[/tex], which does not include values outside of this interval.
Therefore, the range of [tex]\( y = \cos(x + 1) \)[/tex] is [tex]\([-1, 1]\)[/tex].
Option D: [tex]\( y = \cot(2x) - 1 \)[/tex]
Finally, recall that [tex]\( \cot(x) = \frac{\cos(x)}{\sin(x)} \)[/tex] and its range is [tex]\((-\infty, \infty)\)[/tex] because [tex]\( \cot(x) \)[/tex] can take any real value. Subtracting 1 does not change the fact that the range will remain all real numbers:
The range of [tex]\( y = \cot(2x) - 1 \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex].
Conclusion:
From the analysis above, the function that has the range [tex]\( (-\infty, -2] \cup [0, \infty) \)[/tex] is:
A. [tex]\( y = \csc(x) - 1 \)[/tex]
Option A: [tex]\( y = \csc(x) - 1 \)[/tex]
First, recall that [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex], which makes the range of [tex]\( \csc(x) \)[/tex] be [tex]\((-\infty, -1] \cup [1, \infty)\)[/tex]. Now, if we subtract 1 from [tex]\( \csc(x) \)[/tex]:
- For [tex]\( \csc(x) \)[/tex] values in [tex]\((-\infty, -1]\)[/tex]: [tex]\( \csc(x) - 1 \leq -2 \)[/tex]
- For [tex]\( \csc(x) \)[/tex] values in [tex]\([1, \infty)\)[/tex]: [tex]\( \csc(x) - 1 \geq 0 \)[/tex]
Thus, the range of [tex]\( y = \csc(x) - 1 \)[/tex] is [tex]\((-\infty, -2] \cup [0, \infty)\)[/tex].
Option B: [tex]\( y = \sec(x) + 1 \)[/tex]
Next, note that [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex], which makes the range of [tex]\( \sec(x) \)[/tex] be [tex]\((-\infty, -1] \cup [1, \infty)\)[/tex]. Adding 1 to [tex]\( \sec(x) \)[/tex]:
- For [tex]\( \sec(x) \)[/tex] values in [tex]\((-\infty, -1]\)[/tex]: [tex]\( \sec(x) + 1 \leq 0 \)[/tex]
- For [tex]\( \sec(x) \)[/tex] values in [tex]\([1, \infty)\)[/tex]: [tex]\( \sec(x) + 1 \geq 2 \)[/tex]
So, the range of [tex]\( y = \sec(x) + 1 \)[/tex] is [tex]\((-\infty, 0] \cup [2, \infty)\)[/tex], which does not match the required range.
Option C: [tex]\( y = \cos(x + 1) \)[/tex]
For [tex]\( y = \cos(x + 1) \)[/tex], the function essentially has the same range as [tex]\( \cos(x) \)[/tex] since horizontal shifts do not affect the range of cosine. The range of [tex]\( \cos(x) \)[/tex] is [tex]\([-1, 1]\)[/tex], which does not include values outside of this interval.
Therefore, the range of [tex]\( y = \cos(x + 1) \)[/tex] is [tex]\([-1, 1]\)[/tex].
Option D: [tex]\( y = \cot(2x) - 1 \)[/tex]
Finally, recall that [tex]\( \cot(x) = \frac{\cos(x)}{\sin(x)} \)[/tex] and its range is [tex]\((-\infty, \infty)\)[/tex] because [tex]\( \cot(x) \)[/tex] can take any real value. Subtracting 1 does not change the fact that the range will remain all real numbers:
The range of [tex]\( y = \cot(2x) - 1 \)[/tex] is still [tex]\((-\infty, \infty)\)[/tex].
Conclusion:
From the analysis above, the function that has the range [tex]\( (-\infty, -2] \cup [0, \infty) \)[/tex] is:
A. [tex]\( y = \csc(x) - 1 \)[/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.
