If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x
Practice Questions
Q1
If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x = 0, find k.
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Questions & Step-by-Step Solutions
If f(x) = { x^2 + 1, x < 0; kx + 2, x = 0; 3 - x, x > 0 is continuous at x = 0, find k.
Step 1: Understand that the function f(x) is defined in three parts based on the value of x: for x < 0, for x = 0, and for x > 0.
Step 2: Identify the part of the function that applies when x approaches 0 from the left (x < 0). This part is f(x) = x^2 + 1.
Step 3: Calculate the limit of f(x) as x approaches 0 from the left: f(0-) = 0^2 + 1 = 1.
Step 4: Identify the part of the function that applies when x = 0. This part is f(0) = kx + 2. Since x = 0, this simplifies to f(0) = 2.
Step 5: Identify the part of the function that applies when x approaches 0 from the right (x > 0). This part is f(x) = 3 - x.
Step 6: Calculate the limit of f(x) as x approaches 0 from the right: f(0+) = 3 - 0 = 3.
Step 7: For the function to be continuous at x = 0, the left limit (1) must equal the value at x = 0 (2) and the right limit (3).
Step 8: Set the left limit equal to the value at x = 0: 1 = 2. This is not true, so we need to adjust k to make it true.
Step 9: Since we need the left limit (1) to equal the value at x = 0 (2), we find that k must be 1 to satisfy the condition.
Step 10: Therefore, the value of k that makes the function continuous at x = 0 is k = 1.
Continuity of Functions – The question tests the understanding of continuity at a point, specifically how to ensure that the left-hand limit, right-hand limit, and the function value at that point are equal.
Piecewise Functions – The question involves analyzing a piecewise function, which requires careful consideration of the different expressions for different intervals of x.
