<p>If \(f(x+1) - f(x) = e^x\), then \(f(2) - f(0)\) equals</p>
Step-by-Step Solution
Key Concept: Use the functional equation iteratively: f(2) - f(0) = [f(2) - f(1)] + [f(1) - f(0)] = e^1 + e^0. The key is telescoping the differences using the given relation at different values of x.
<p><strong>Step 1:</strong> Write the functional equation f(x+1) - f(x) = e^x</p><p><strong>Step 2:</strong> Apply at x = 0: f(1) - f(0) = e^0 = 1</p><p><strong>Step 3:</strong> Apply at x = 1: f(2) - f(1) = e^1 = e</p><p><strong>Step 4:</strong> Add both equations: [f(1) - f(0)] + [f(2) - f(1)] = 1 + e</p><p><strong>Step 5:</strong> The middle terms cancel (telescoping): f(2) - f(0) = 1 + e</p><p>∴ Answer: C (which is e + 1)</p>
Correct Answer: C