Which pair of functions have the same domain?

Question

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

Answer 1

To determine which pairs of functions have the same domain, we need to examine the restrictions for each function.

1. Cosecant \( f(x) = \csc(x) \): The function is defined wherever \( \sin(x) \neq 0 \). This means it is undefined at \( x = n\pi \) for any integer \( n \).

   Tangent \( g(x) = \tan(x) \): The function is defined wherever \( \cos(x) \neq 0 \). This means it is undefined at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).

   Domain: Different

2. Cosine \( f(x) = \cos(x) \): This function is defined for all \( x \).

   Secant \( g(x) = \sec(x) \): The function is defined wherever \( \cos(x) \neq 0 \), which means it is undefined at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).

   Domain: Different

3. Sine \( f(x) = \sin(x) \): This function is defined for all \( x \).

   Cosine \( g(x) = \cos(x) \): This function is also defined for all \( x \).

   Domain: Same

4. Secant \( f(x) = \sec(x) \): As discussed earlier, it is undefined at \( x = \frac{\pi}{2} + n\pi \).

   Cotangent \( g(x) = \cot(x) \): This function is undefined wherever \( \sin(x) = 0 \), specifically at \( x = n\pi \) for any integer \( n \).

   Domain: Different

Based on this analysis, the pair of functions that have the same domain is:

f(x) = sine x and g(x) = cosine x.