<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: Use substitution strategically to create a system of equations. Replace x with a transformed value that maps back to the original, creating two equations you can solve simultaneously for f(x).
<p><strong>Step 1:</strong> Write the original equation:</p><p>3f(x) + 2f((x+59)/(x-1)) = 10x + 30 ... (1)</p><p><strong>Step 2:</strong> Let u = (x+59)/(x-1). Find what x becomes when we substitute u into the argument:</p><p>If y = (x+59)/(x-1), then x = (y+59)/(y-1) gives us the inverse relationship. Substituting x → (x+59)/(x-1) in equation (1):</p><p>3f((x+59)/(x-1)) + 2f(x) = 10·(x+59)/(x-1) + 30 ... (2)</p><p><strong>Step 3:</strong> Verify the transformation cycles correctly. When x=7: (7+59)/(7-1) = 66/6 = 11</p><p>When x=11: (11+59)/(11-1) = 70/10 = 7 ✓ (cycles back)</p><p><strong>Step 4:</strong> Create system using x=7 in equation (1):</p><p>3f(7) + 2f(11) = 70 + 30 = 100 ... (A)</p><p><strong>Step 5:</strong> Create second equation using x=11 in equation (1):</p><p>3f(11) + 2f(7) = 110 + 30 = 140 ... (B)</p><p><strong>Step 6:</strong> Solve the system. Multiply (A) by 3: 9f(7) + 6f(11) = 300</p><p>Multiply (B) by 2: 6f(11) + 4f(7) = 280</p><p>Subtract: 5f(7) = 20</p><p><strong>∴ Answer: f(7) = 4 (Option A)</strong></p>
Correct Answer: A