<p>If \( f(x) \) is an odd function, then \( f(0) \) equals:</p>
Step-by-Step Solution
Key Concept: For an odd function, f(-x) = -f(x) must hold for all x in the domain. Setting x = 0 immediately gives f(0) = -f(0), which forces f(0) = 0.
<p><strong>Step 1:</strong> Recall that f(x) is odd if and only if f(-x) = -f(x) for all x in the domain.</p><p><strong>Step 2:</strong> Apply this property at x = 0: f(-0) = -f(0)</p><p><strong>Step 3:</strong> Simplify: f(0) = -f(0)</p><p><strong>Step 4:</strong> This equation is satisfied only when 2f(0) = 0, so f(0) = 0</p><p><strong>Insight:</strong> The origin (0,0) must always lie on the graph of any odd function, making this a fundamental property.</p><p>∴ Answer: <strong>f(0) = 0</strong></p>
Correct Answer: C