Question

what is the domain of $f(x) = \cos(x)$?
\bigcirc the set of real numbers $-2\pi \leq x \leq 2\pi$
\bigcirc the set of real numbers $-1 \leq x \leq 1$
\bigcirc the set of real numbers $0 \leq x \leq 2\pi$
\bigcirc the set of all real numbers

Explanation:

Step1: Recall domain of cosine

The cosine function \( f(x)=\cos(x) \) is defined for all real numbers. There are no restrictions on the input \( x \) (in terms of real numbers) for the cosine function.

Step2: Analyze options

• Option 1: \( -2\pi\leq x\leq 2\pi \) is a restricted interval, not the full domain.
• Option 2: \( -1\leq x\leq 1 \) is the range of \( \cos(x) \), not the domain.
• Option 3: \( 0\leq x\leq 2\pi \) is a period of the cosine function, not the full domain.
• Option 4: The set of all real numbers matches the domain of \( \cos(x) \).

Answer:

the set of all real numbers