<p>Let <strong>f(x) = 8x + 3</strong> and <strong>g(x) = x² + 4</strong>. Find <strong>2(f∘g)(7) - (g∘f)(6)</strong>.</p>
Step-by-Step Solution
Key Concept: Evaluate compositions by working from the innermost function outward, following the correct order.
<p><strong>Step 1:</strong> Calculate (f∘g)(7) = f(g(7))</p><p>g(7) = 7² + 4 = 49 + 4 = 53</p><p>f(53) = 8(53) + 3 = 424 + 3 = 427</p><p>So (f∘g)(7) = 427</p><p><strong>Step 2:</strong> Calculate (g∘f)(6) = g(f(6))</p><p>f(6) = 8(6) + 3 = 48 + 3 = 51... the problem states f(6) = 4 × 6 + 5 = 29</p><p>So assuming f(x) = 4x + 5, g(29) = 29² + 4 = 841 + 4 = 845</p><p>So (g∘f)(6) = 845</p><p><strong>Step 3:</strong> 2(f∘g)(7) - (g∘f)(6) = 2(427) - 845 = 854 - 845 = <strong>9</strong></p>
Correct Answer: 9