To determine the domain of the function \( f(x) = \cos(x) \), we need to identify the set of values for \( x \) for which the function is defined.
The cosine function, \(\cos(x)\), is a trigonometric function that relates to the coordinates of points on the unit circle. Since the unit circle encompasses all possible angles, the cosine function can take an input of any real number and still produce a valid output (which will be a value between -1 and 1 inclusive).
Here is the step-by-step reasoning:
1. Understanding the function: \( f(x) = \cos(x) \) is defined for any real number \( x \). In other words, there is no restriction on \( x \) that would make the function undefined.
2. Domain Analysis:
- The cosine function does not have any discontinuities. It is continuous and periodic for all real numbers.
- It does not have any vertical asymptotes, holes, or breaks. Every real number \( x \) corresponds to a unique value of \(\cos(x)\).
3. Reaffirming the knowledge:
- The cosine of any real number, whether positive, negative, or zero, is always defined.
- The periodic nature of the cosine function means it repeats its values in regular intervals (specifically, every \(2\pi\) radians), but this repetition does not limit its domain.
Given these points, we conclude that the domain of \( f(x) = \cos(x) \) is the set of all real numbers.
Therefore, the correct answer is:
The set of all real numbers.
