<p>Given <em>f</em>(<em>xy</em>) = <em>f</em>(<em>x</em>) · <em>f</em>(<em>y</em>), for all \(x, y \in [0,1]\). Since \(f(0) \neq 0\), then \(f(0) = 1\). So \(f(x) = 1\). Find \(y\left(\frac{1}{4}\right) + y\left(\frac{3}{4}\right)\).</p>
Step-by-Step Solution
Key Concept: A functional equation f(xy) = f(x)·f(y) with f(0) ≠ 0 forces f to be constant. Setting y = 0 gives f(0) = f(x)·f(0), so if f(0) ≠ 0, then f(x) = 1 for all x. The notation 'y' in the final question appears to be a typo for 'f', so we evaluate f(1/4) + f(3/4) = 1 + 1 = 2.
<p><strong>Step 1: Use the functional equation with y = 0</strong></p><p>Given: f(xy) = f(x)·f(y) for all x, y ∈ [0,1]</p><p>Set y = 0: f(x·0) = f(x)·f(0)</p><p>This gives: f(0) = f(x)·f(0)</p><p><strong>Step 2: Solve for f(x)</strong></p><p>Since f(0) ≠ 0, we can divide both sides by f(0):</p><p>1 = f(x)</p><p>Therefore: f(x) = 1 for all x ∈ [0,1]</p><p><strong>Step 3: Evaluate the sum</strong></p><p>Interpreting y(·) as f(·) (correcting the typo):</p><p>y(1/4) + y(3/4) = f(1/4) + f(3/4) = 1 + 1 = <strong>2</strong></p><p>∴ Answer: C</p>
Correct Answer: C