Final answer:
The domain of the function is all real numbers except those that make cos(x) negative, and the range is all real numbers greater than or equal to 0.
Step-by-step explanation:
The domain of a function represents all the possible input values, while the range represents all the possible output values.
For the given function f(x) = x - |2 cos x|, let's start by finding the domain.
The function includes the cosine function, which has a domain of all real numbers. However, the absolute value function restricts the values that can be input into it.
Since the absolute value function always returns a positive value, the only values of x that will affect the output are those that make cos(x) negative. The values of x for which cos(x) is negative can be found by considering the intervals when cos(x) < 0.
Therefore, the domain of the function f(x) = x - |2 cos x| is the set of all real numbers except for those that make cos(x) negative: x ≠ (2n + 1)π, where n is an integer.
In terms of the range, since the cosine function oscillates between -1 and 1, the absolute value function will always yield positive values. Therefore, the range of the function f(x) = x - |2 cos x| is all real numbers greater than or equal to 0.