<p>Let \(f: R \to R\) be continuous function and \(f(x) = f(2x)\) is true \(\forall\, x \in R\) and \(f(1) = 3\), then the value of \(\displaystyle\int_{-1}^{1} f(f(x))\, dx\) is equal to:</p>
Step-by-Step Solution
Key Concept: If f(x) = f(2x) for all x ∈ ℝ, then f is constant. Substituting x → x/2 repeatedly shows f(x) = f(x/2ⁿ) for all n, and as n → ∞, the argument → 0, so f(x) = f(0) for all x. Since f(1) = 3, we have f(x) = 3 everywhere.
<p><strong>Step 1:</strong> Analyze the functional equation f(x) = f(2x).</p><p>Replacing x with x/2: f(x/2) = f(x). Therefore f(x) = f(x/2) = f(x/4) = f(x/8) = ... = f(x/2ⁿ).</p><p><strong>Step 2:</strong> Take the limit as n → ∞.</p><p>As n → ∞, x/2ⁿ → 0. By continuity of f: f(x) = lim(n→∞) f(x/2ⁿ) = f(0).</p><p>Thus f(x) = f(0) for all x ∈ ℝ, meaning f is constant.</p><p><strong>Step 3:</strong> Find the constant value.</p><p>Since f(1) = 3 and f is constant, f(x) = 3 for all x ∈ ℝ.</p><p><strong>Step 4:</strong> Evaluate f(f(x)).</p><p>f(f(x)) = f(3) = 3.</p><p><strong>Step 5:</strong> Compute the definite integral.</p><p>∫₋₁¹ f(f(x)) dx = ∫₋₁¹ 3 dx = 3[x]₋₁¹ = 3(1 - (-1)) = 3(2) = 6.</p><p>∴ Answer: C (or 6)</p>
Correct Answer: C