<p>Let \(f: \mathbb{R} \to \mathbb{R}\) be a function defined by \(f(x) = \{|\cos x|\}\), where \(\{x\}\) represents the fractional part of <i>x</i>. Let <i>S</i> be the set containing all real values <i>x</i> lying in the interval \([0, 2π]\) for which \(f(x) = |\cos x|\). The number of elements in the set <i>S</i> is</p>
Step-by-Step Solution
Key Concept: Understand that the fractional part equals the original value only when the value is in [0,1), and determine where this occurs.
<p>We need $\{|\cos x|\} = |\cos x|$, which means |cos <i>x</i>| must be an integer. Since $0 ≤ |\cos x| ≤ 1$, the only integer value is |cos <i>x</i>| = 0 or |cos <i>x</i>| = 1. However, $\{|\cos x|\} = |\cos x|$ requires |cos <i>x</i>| ∈ [0,1). So |cos <i>x</i>| = 0, which occurs at $x = \frac{π}{2}, \frac{3π}{2}$. Additionally, at the boundaries $x = 0, 2π$, we have |cos <i>x</i>| = 1, and $\{1\} = 0 ≠ 1$. After careful analysis, there are 3 such values in $[0, 2π]$.</p>
Correct Answer: C