What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
Practice Questions
1 question
Q1
What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
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Setting 4 = 2 gives a = 1 for continuity.
Questions & Step-by-step Solutions
1 item
Q
Q: What value of a makes the function f(x) = { 4 - x^2, x < 0; ax + 2, x = 0; x + 1, x > 0 continuous at x = 0?
Solution: Setting 4 = 2 gives a = 1 for continuity.
Steps: 10
Step 1: Understand that we want the function f(x) to be continuous at x = 0.
Step 2: Recall that for a function to be continuous at a point, the left-hand limit, right-hand limit, and the function value at that point must all be equal.
Step 3: Identify the function pieces: f(x) = 4 - x^2 for x < 0, f(x) = ax + 2 for x = 0, and f(x) = x + 1 for x > 0.
Step 4: Calculate the left-hand limit as x approaches 0: f(0-) = 4 - (0)^2 = 4.
Step 5: Calculate the right-hand limit as x approaches 0: f(0+) = 0 + 1 = 1.
Step 6: Set the left-hand limit equal to the function value at x = 0: 4 = ax + 2 when x = 0, which simplifies to 4 = 2.
Step 7: Solve for a: Since 4 = 2 is not dependent on a, we need to ensure the left-hand limit equals the right-hand limit.
Step 8: Set the left-hand limit equal to the right-hand limit: 4 = 1, which is not possible, so we need to adjust the function value at x = 0.
Step 9: Set the function value at x = 0 equal to the left-hand limit: ax + 2 = 4 when x = 0, which simplifies to 2 = 4 - 2.
Step 10: Solve for a: 4 - 2 = 2, so we need a = 1 to make the function continuous.
