Answer :
To determine which of these functions has a range of all real numbers, let's analyze each function one by one.
### A. [tex]\( y = \sec(x) \)[/tex]
The secant function is defined as [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex].
- The cosine function, [tex]\( \cos(x) \)[/tex], ranges between -1 and 1.
- Therefore, [tex]\( \sec(x) \)[/tex] will have vertical asymptotes where [tex]\( \cos(x) = 0 \)[/tex] (i.e., [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer).
- As [tex]\( \cos(x) \)[/tex] approaches 0, [tex]\( \sec(x) \)[/tex] approaches [tex]\( \pm \infty \)[/tex]. However, [tex]\( \sec(x) \)[/tex] only takes values greater than or equal to 1 or less than or equal to -1.
- Thus, [tex]\( y = \sec(x) \)[/tex] does not have the range of all real numbers.
### B. [tex]\( y = \tan(x) \)[/tex]
The tangent function is defined as [tex]\( \tan(x) = \frac{\sin(x)}{\cos(x)} \)[/tex].
- The cosine function, [tex]\( \cos(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex]. At these points, [tex]\( \tan(x) \)[/tex] has vertical asymptotes.
- Between these asymptotes, [tex]\( \tan(x) \)[/tex] takes all real values from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Thus, the range of [tex]\( y = \tan(x) \)[/tex] is all real numbers.
### C. [tex]\( y = \cot(x) \)[/tex]
The cotangent function is defined as [tex]\( \cot(x) = \frac{\cos(x)}{\sin(x)} \)[/tex].
- The sine function, [tex]\( \sin(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer. At these points, [tex]\( \cot(x) \)[/tex] has vertical asymptotes.
- Between these asymptotes, [tex]\( \cot(x) \)[/tex] spans all real values from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
- However, unlike the tangent function, [tex]\( \cot(x) \)[/tex] does not achieve all values between these asymptotes as [tex]\( x \to n\pi \)[/tex]. Specifically, it does not take all possible real values within each period and approaches only finite limits around each [tex]\( n\pi \)[/tex].
Thus, [tex]\( y = \cot(x) \)[/tex] does not have the range of all real numbers.
### D. [tex]\( y = \csc(x) \)[/tex]
The cosecant function is defined as [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex].
- The sine function, [tex]\( \sin(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer. At these points, [tex]\( \csc(x) \)[/tex] has vertical asymptotes.
- As [tex]\( \sin(x) \)[/tex] approaches 0, [tex]\( \csc(x) \)[/tex] approaches [tex]\( \pm \infty \)[/tex]. However, [tex]\( \csc(x) \)[/tex] will only take values greater than or equal to 1 or less than or equal to -1.
Thus, [tex]\( y = \csc(x) \)[/tex] does not have the range of all real numbers.
### Conclusion
From the analysis, the function that has the range of all real numbers is:
[tex]\[ \boxed{B. \, y = \tan(x)} \][/tex]
### A. [tex]\( y = \sec(x) \)[/tex]
The secant function is defined as [tex]\( \sec(x) = \frac{1}{\cos(x)} \)[/tex].
- The cosine function, [tex]\( \cos(x) \)[/tex], ranges between -1 and 1.
- Therefore, [tex]\( \sec(x) \)[/tex] will have vertical asymptotes where [tex]\( \cos(x) = 0 \)[/tex] (i.e., [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer).
- As [tex]\( \cos(x) \)[/tex] approaches 0, [tex]\( \sec(x) \)[/tex] approaches [tex]\( \pm \infty \)[/tex]. However, [tex]\( \sec(x) \)[/tex] only takes values greater than or equal to 1 or less than or equal to -1.
- Thus, [tex]\( y = \sec(x) \)[/tex] does not have the range of all real numbers.
### B. [tex]\( y = \tan(x) \)[/tex]
The tangent function is defined as [tex]\( \tan(x) = \frac{\sin(x)}{\cos(x)} \)[/tex].
- The cosine function, [tex]\( \cos(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = \frac{\pi}{2} + n\pi \)[/tex]. At these points, [tex]\( \tan(x) \)[/tex] has vertical asymptotes.
- Between these asymptotes, [tex]\( \tan(x) \)[/tex] takes all real values from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
Thus, the range of [tex]\( y = \tan(x) \)[/tex] is all real numbers.
### C. [tex]\( y = \cot(x) \)[/tex]
The cotangent function is defined as [tex]\( \cot(x) = \frac{\cos(x)}{\sin(x)} \)[/tex].
- The sine function, [tex]\( \sin(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer. At these points, [tex]\( \cot(x) \)[/tex] has vertical asymptotes.
- Between these asymptotes, [tex]\( \cot(x) \)[/tex] spans all real values from [tex]\( -\infty \)[/tex] to [tex]\( \infty \)[/tex].
- However, unlike the tangent function, [tex]\( \cot(x) \)[/tex] does not achieve all values between these asymptotes as [tex]\( x \to n\pi \)[/tex]. Specifically, it does not take all possible real values within each period and approaches only finite limits around each [tex]\( n\pi \)[/tex].
Thus, [tex]\( y = \cot(x) \)[/tex] does not have the range of all real numbers.
### D. [tex]\( y = \csc(x) \)[/tex]
The cosecant function is defined as [tex]\( \csc(x) = \frac{1}{\sin(x)} \)[/tex].
- The sine function, [tex]\( \sin(x) \)[/tex], ranges between -1 and 1 and is zero at [tex]\( x = n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer. At these points, [tex]\( \csc(x) \)[/tex] has vertical asymptotes.
- As [tex]\( \sin(x) \)[/tex] approaches 0, [tex]\( \csc(x) \)[/tex] approaches [tex]\( \pm \infty \)[/tex]. However, [tex]\( \csc(x) \)[/tex] will only take values greater than or equal to 1 or less than or equal to -1.
Thus, [tex]\( y = \csc(x) \)[/tex] does not have the range of all real numbers.
### Conclusion
From the analysis, the function that has the range of all real numbers is:
[tex]\[ \boxed{B. \, y = \tan(x)} \][/tex]
