<p>Let \( f(x) \) be defined as \( f(x) = kx + 2 \) for \( x \leq -1 \) and \( f(x) = 2x + 3 \) for \( x > -1 \). If \( f \) is continuous at \( x = -1 \), then \( k \) equals</p>
Step-by-Step Solution
Key Concept: For continuity at a point, the left-hand limit, right-hand limit, and function value must all be equal. Use this condition at x = -1 to find k by equating the two piece-wise expressions.
<p><strong>Step 1:</strong> Identify which pieces apply at x = -1. Since x ≤ -1, we use f(x) = kx + 2 at x = -1.</p><p><strong>Step 2:</strong> Calculate the left-hand limit as x → -1⁻ using the first piece: <br>lim(x→-1⁻) f(x) = k(-1) + 2 = -k + 2</p><p><strong>Step 3:</strong> Calculate the right-hand limit as x → -1⁺ using the second piece: <br>lim(x→-1⁺) f(x) = 2(-1) + 3 = 1</p><p><strong>Step 4:</strong> For continuity at x = -1, the left limit must equal the right limit:<br>-k + 2 = 1<br>-k = -1<br>k = 1</p><p><strong>Verification:</strong> f(-1) = 1(-1) + 2 = 1, which matches both limits.</p><p>∴ Answer: B (k = 1)</p>
Correct Answer: B