<p>The function \(f(x)\) satisfies the functional equation \(3f(x) + 2f\!\left(\dfrac{x+59}{x-1}\right) = 10x + 30\) for all real \(x \neq 1\). The value of \(f(7)\) is:</p>
Step-by-Step Solution
Key Concept: Create a system of equations by substituting strategically chosen values into the functional equation. Find a value that maps back to itself under the transformation x → (x+59)/(x-1), enabling you to solve for f(x) explicitly.
<p><strong>Step 1:</strong> Let g(x) = (x+59)/(x-1). Verify that g(g(x)) = x by computing: g(g(x)) = g((x+59)/(x-1)) = [((x+59)/(x-1)) + 59]/[((x+59)/(x-1)) - 1] = [(x+59+59(x-1))/(x-1)]/[(x+59-(x-1))/(x-1)] = [60x]/[60] = x ✓</p><p><strong>Step 2:</strong> Write the original equation at x: 3f(x) + 2f(g(x)) = 10x + 30 ... (1)</p><p><strong>Step 3:</strong> Replace x with g(x) in the equation. Since g(g(x)) = x, we get: 3f(g(x)) + 2f(x) = 10g(x) + 30 = 10·(x+59)/(x-1) + 30 ... (2)</p><p><strong>Step 4:</strong> Simplify equation (2): 3f(g(x)) + 2f(x) = (10x + 590 + 30(x-1))/(x-1) = (40x + 560)/(x-1)</p><p><strong>Step 5:</strong> Solve the system. Multiply (1) by 3: 9f(x) + 6f(g(x)) = 30x + 90. Multiply (2) by 2: 6f(g(x)) + 4f(x) = (80x + 1120)/(x-1). Subtract: 5f(x) = 30x + 90 - (80x + 1120)/(x-1) = [30x(x-1) + 90(x-1) - 80x - 1120]/(x-1) = [30x² + 10x - 1210]/(x-1)</p><p><strong>Step 6:</strong> Therefore f(x) = [6x² + 2x - 242]/(x-1). At x = 7: f(7) = [6(49) + 2(7) - 242]/6 = [294 + 14 - 242]/6 = 66/6 = 11</p><p>∴ Answer: A (f(7) = 11)</p>
Correct Answer: A