Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To determine which of the given functions have a range that covers all real numbers, let's analyze each function individually.
### A. [tex]\( y = \tan x \)[/tex]
The tangent function, [tex]\(\tan x\)[/tex], is defined as [tex]\(\sin x / \cos x\)[/tex]. This function experiences vertical asymptotes where [tex]\(\cos x = 0\)[/tex], specifically at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] for any integer [tex]\( k \)[/tex]. However, between these asymptotes, [tex]\(\tan x\)[/tex] will take on every possible real value because it increases and decreases without bound within each interval [tex]\((\frac{\pi}{2} + k\pi, \frac{\pi}{2} + (k+1)\pi)\)[/tex]. Therefore, the range of the tangent function is all real numbers.
Conclusion: [tex]\( y = \tan x \)[/tex] has a range of all real numbers.
### B. [tex]\( y = \csc x \)[/tex]
The cosecant function, [tex]\(\csc x\)[/tex], is the reciprocal of the sine function, defined as [tex]\( 1 / \sin x \)[/tex]. The sine function ranges between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], but [tex]\(\sin x\)[/tex] cannot be zero since the reciprocal function [tex]\(\csc x\)[/tex] would be undefined at those points. Consequently, the values that [tex]\(\csc x\)[/tex] can take are those values where [tex]\(|\sin x| \geq 1\)[/tex]. Therefore, the range of the cosecant function is [tex]\( (-\infty, -1] \cup [1, \infty) \)[/tex].
Conclusion: [tex]\( y = \csc x \)[/tex] does not have a range of all real numbers.
### C. [tex]\( y = \sec x \)[/tex]
The secant function, [tex]\(\sec x\)[/tex], is the reciprocal of the cosine function, defined as [tex]\( 1 / \cos x \)[/tex]. The cosine function ranges between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], but [tex]\(\cos x\)[/tex] cannot be zero since the reciprocal function [tex]\(\sec x\)[/tex] would be undefined at those points. Consequently, the values that [tex]\(\sec x\)[/tex] can take are those values where [tex]\(|\cos x| \geq 1\)[/tex]. Therefore, the range of the secant function is [tex]\( (-\infty, -1] \cup [1, \infty) \)[/tex].
Conclusion: [tex]\( y = \sec x \)[/tex] does not have a range of all real numbers.
### D. [tex]\( y = \cot x \)[/tex]
The cotangent function, [tex]\(\cot x\)[/tex], is defined as [tex]\(\cos x / \sin x\)[/tex]. This function experiences vertical asymptotes where [tex]\(\sin x = 0\)[/tex], specifically at [tex]\( x = k\pi \)[/tex] for any integer [tex]\( k \)[/tex]. However, between these asymptotes, [tex]\(\cot x\)[/tex] will take on every possible real value because it increases and decreases without bound within each interval [tex]\((k\pi, (k+1)\pi)\)[/tex]. Therefore, the range of the cotangent function is all real numbers.
Conclusion: [tex]\( y = \cot x \)[/tex] does not have a range of all real numbers.
### Summary
From our analysis:
- [tex]\( y = \tan x \)[/tex] has a range of all real numbers.
- [tex]\( y = \csc x \)[/tex], [tex]\( y = \sec x \)[/tex], and [tex]\( y = \cot x \)[/tex] do not have ranges that cover all real numbers.
Thus, the function that has a range of all real numbers is:
[tex]\[ \boxed{A. \ y = \tan x} \][/tex]
### A. [tex]\( y = \tan x \)[/tex]
The tangent function, [tex]\(\tan x\)[/tex], is defined as [tex]\(\sin x / \cos x\)[/tex]. This function experiences vertical asymptotes where [tex]\(\cos x = 0\)[/tex], specifically at [tex]\( x = \frac{\pi}{2} + k\pi \)[/tex] for any integer [tex]\( k \)[/tex]. However, between these asymptotes, [tex]\(\tan x\)[/tex] will take on every possible real value because it increases and decreases without bound within each interval [tex]\((\frac{\pi}{2} + k\pi, \frac{\pi}{2} + (k+1)\pi)\)[/tex]. Therefore, the range of the tangent function is all real numbers.
Conclusion: [tex]\( y = \tan x \)[/tex] has a range of all real numbers.
### B. [tex]\( y = \csc x \)[/tex]
The cosecant function, [tex]\(\csc x\)[/tex], is the reciprocal of the sine function, defined as [tex]\( 1 / \sin x \)[/tex]. The sine function ranges between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], but [tex]\(\sin x\)[/tex] cannot be zero since the reciprocal function [tex]\(\csc x\)[/tex] would be undefined at those points. Consequently, the values that [tex]\(\csc x\)[/tex] can take are those values where [tex]\(|\sin x| \geq 1\)[/tex]. Therefore, the range of the cosecant function is [tex]\( (-\infty, -1] \cup [1, \infty) \)[/tex].
Conclusion: [tex]\( y = \csc x \)[/tex] does not have a range of all real numbers.
### C. [tex]\( y = \sec x \)[/tex]
The secant function, [tex]\(\sec x\)[/tex], is the reciprocal of the cosine function, defined as [tex]\( 1 / \cos x \)[/tex]. The cosine function ranges between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex], but [tex]\(\cos x\)[/tex] cannot be zero since the reciprocal function [tex]\(\sec x\)[/tex] would be undefined at those points. Consequently, the values that [tex]\(\sec x\)[/tex] can take are those values where [tex]\(|\cos x| \geq 1\)[/tex]. Therefore, the range of the secant function is [tex]\( (-\infty, -1] \cup [1, \infty) \)[/tex].
Conclusion: [tex]\( y = \sec x \)[/tex] does not have a range of all real numbers.
### D. [tex]\( y = \cot x \)[/tex]
The cotangent function, [tex]\(\cot x\)[/tex], is defined as [tex]\(\cos x / \sin x\)[/tex]. This function experiences vertical asymptotes where [tex]\(\sin x = 0\)[/tex], specifically at [tex]\( x = k\pi \)[/tex] for any integer [tex]\( k \)[/tex]. However, between these asymptotes, [tex]\(\cot x\)[/tex] will take on every possible real value because it increases and decreases without bound within each interval [tex]\((k\pi, (k+1)\pi)\)[/tex]. Therefore, the range of the cotangent function is all real numbers.
Conclusion: [tex]\( y = \cot x \)[/tex] does not have a range of all real numbers.
### Summary
From our analysis:
- [tex]\( y = \tan x \)[/tex] has a range of all real numbers.
- [tex]\( y = \csc x \)[/tex], [tex]\( y = \sec x \)[/tex], and [tex]\( y = \cot x \)[/tex] do not have ranges that cover all real numbers.
Thus, the function that has a range of all real numbers is:
[tex]\[ \boxed{A. \ y = \tan x} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
