If f(x) = ax + b and f(f(f(x))) = 27x + 13 where a and b are real numbers, then -
Step-by-Step Solution
Key Concept: Composition of linear functions produces another linear function; by composing f(x) = ax + b with itself three times and comparing coefficients with 27x + 13, we can determine a and b uniquely.
<p><strong>Step 1:</strong> Find f(f(x)).</p><p>Given f(x) = ax + b, we compute:</p><p>f(f(x)) = f(ax + b) = a(ax + b) + b = a²x + ab + b</p><p><strong>Step 2:</strong> Find f(f(f(x))).</p><p>f(f(f(x))) = f(a²x + ab + b) = a(a²x + ab + b) + b</p><p>= a³x + a²b + ab + b</p><p>= a³x + b(a² + a + 1)</p><p><strong>Step 3:</strong> Compare with the given condition 27x + 13.</p><p>We have f(f(f(x))) = a³x + b(a² + a + 1) = 27x + 13</p><p>Comparing coefficients:</p><p>• Coefficient of x: a³ = 27 ⟹ a = 3</p><p>• Constant term: b(a² + a + 1) = 13</p><p><strong>Step 4:</strong> Substitute a = 3 into the constant term equation.</p><p>b(3² + 3 + 1) = 13</p><p>b(9 + 3 + 1) = 13</p><p>b(13) = 13</p><p>b = 1</p><p><strong>Step 5:</strong> Verify and find a + b.</p><p>With a = 3 and b = 1:</p><p>• f(x) = 3x + 1</p><p>• f(f(x)) = 3(3x + 1) + 1 = 9x + 4</p><p>• f(f(f(x))) = 3(9x + 4) + 1 = 27x + 13 ✓</p><p>Therefore: a + b = 3 + 1 = 4</p><p><strong>Checking the options:</strong></p><p>• Option A: a + b = 3 is false (we found 4)</p><p>• Option B: a + b = 4 is true ✓</p><p>• Option C: f(x) = 3 is false (f(x) = 3x + 1)</p><p>• Option D: f'(x) = -3 is false (f'(x) = 3)</p><p><strong>∴ Answer: B</strong></p>
Correct Answer: B