Determine the value of p for which the function f(x) = { x^2 + p, x < 0; 1, x
Practice Questions
Q1
Determine the value of p for which the function f(x) = { x^2 + p, x < 0; 1, x = 0; 2x + p, x > 0 is continuous at x = 0.
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Questions & Step-by-Step Solutions
Determine the value of p for which the function f(x) = { x^2 + p, x < 0; 1, x = 0; 2x + p, x > 0 is continuous at x = 0.
Step 1: Understand that the function f(x) has different expressions based on the value of x: it has one expression for x < 0, one for x = 0, and another for x > 0.
Step 2: Identify the value of the function at x = 0. According to the function definition, f(0) = 1.
Step 3: For the function to be continuous at x = 0, the limit of f(x) as x approaches 0 from the left (x < 0) must equal f(0) and the limit as x approaches 0 from the right (x > 0) must also equal f(0).
Step 4: Calculate the limit of f(x) as x approaches 0 from the left (x < 0). This is given by the expression x^2 + p. As x approaches 0, this limit becomes 0^2 + p = p.
Step 5: Calculate the limit of f(x) as x approaches 0 from the right (x > 0). This is given by the expression 2x + p. As x approaches 0, this limit becomes 2(0) + p = p.
Step 6: Set the limits equal to f(0) for continuity. This means we need p = 1 from the left limit and p = 1 from the right limit.
Step 7: Conclude that for the function to be continuous at x = 0, p must equal 1.
Continuity of Piecewise Functions – The question tests the understanding of how to ensure continuity at a point for piecewise-defined functions by matching the limits from both sides to the function value at that point.
Limit Evaluation – It requires evaluating the left-hand limit and right-hand limit as x approaches 0 and ensuring they equal the function value at x = 0.
