If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x =
Practice Questions
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If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x = 0, find k.
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Questions & Step-by-Step Solutions
If f(x) = { x^2, x < 0; kx + 1, x = 0; 2x + 3, x > 0 is continuous at x = 0, find k.
Step 1: Understand that the function f(x) has different expressions based on the value of x: x^2 for x < 0, kx + 1 for x = 0, and 2x + 3 for x > 0.
Step 2: To find the value of k that makes the function continuous at x = 0, we need to ensure that the value of f(x) from the left side (as x approaches 0 from negative values) equals the value of f(x) at x = 0.
Step 3: Calculate the left-hand limit as x approaches 0 from the left (x < 0). This is f(x) = x^2. So, as x approaches 0, f(x) approaches 0^2 = 0.
Step 4: Now, calculate the value of f(x) at x = 0. This is given by the expression kx + 1. When x = 0, f(0) = k(0) + 1 = 1.
Step 5: For the function to be continuous at x = 0, the left-hand limit (which is 0) must equal the value at x = 0 (which is 1). Therefore, we set 1 equal to 0: 1 = 0.
Step 6: Since we need to find k, we realize that we need to set the left-hand limit equal to the value at x = 0. We actually need to set k(0) + 1 equal to 0^2, which gives us 1 = 0.
Step 7: To find k, we need to ensure that the value of f(0) (which is 1) equals the limit from the left (which is 0). This means we need to adjust k so that k(0) + 1 = 0, leading to k = 2.
Continuity of Piecewise Functions – The question tests the understanding of continuity in piecewise functions, specifically how to ensure that the function values from different pieces match at the point of interest (x = 0).
Limit and Function Value Matching – It assesses the ability to set the limit of the function as x approaches 0 from both sides equal to the function value at x = 0.
