Question

Which pair of functions have the same domain?
(1 point)
Responses

f left parenthesis x right parenthesis equals cosecant x and g left parenthesis x right parenthesis equals tangent x
Image with alt text: f left parenthesis x right parenthesis equals cosecant x and Image with alt text: g left parenthesis x right parenthesis equals tangent x

f left parenthesis x right parenthesis equals cosine x and f left parenthesis x right parenthesis equals secant x
Image with alt text: f left parenthesis x right parenthesis equals cosine x and Image with alt text: f left parenthesis x right parenthesis equals secant x

f left parenthesis x right parenthesis equals sine x and f left parenthesis x right parenthesis equals cosine x
Image with alt text: f left parenthesis x right parenthesis equals sine x and Image with alt text: f left parenthesis x right parenthesis equals cosine x

f left parenthesis x right parenthesis equals secant x and f left parenthesis x right parenthesis equals cotangent x

Answer 1

To determine which pair of functions have the same domain, we need to analyze the domains of the given functions.

1. **f(x) = cosecant(x) and g(x) = tangent(x)**:
- The cosecant function, \( \csc(x) \), is undefined where \( \sin(x) = 0 \), which occurs at \( x = n\pi \), where \( n \) is an integer.
- The tangent function, \( \tan(x) \), is undefined where \( \cos(x) = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
- Therefore, these two functions have different domains.

2. **f(x) = cosine(x) and g(x) = secant(x)**:
- The cosine function, \( \cos(x) \), is defined for all real numbers.
- The secant function, \( \sec(x) \), is undefined where \( \cos(x) = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
- Thus, these two functions also have different domains.

3. **f(x) = sine(x) and g(x) = cosine(x)**:
- Both the sine function, \( \sin(x) \), and the cosine function, \( \cos(x) \), are defined for all real numbers.
- Therefore, they have the same domain.

4. **f(x) = secant(x) and g(x) = cotangent(x)**:
- The secant function, \( \sec(x) \), is undefined where \( \cos(x) = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
- The cotangent function, \( \cot(x) \), is undefined where \( \sin(x) = 0 \), which occurs at \( x = n\pi \), where \( n \) is an integer.
- Therefore, these two functions also have different domains.

**Conclusion**: The pair that has the same domain is **f(x) = sine(x) and g(x) = cosine(x)**.