Answer :
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]
