The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which

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Question: The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?

Options:

Correct Answer: x = 0

Solution:

To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.

The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?

The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?

Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 for x < 0 and f(x) = 2x + 1 for x >= 0.
Step 2: To check if the function is continuous at x = 0, we need to find f(0).
Step 3: Calculate f(0) using the second part of the function since 0 is greater than or equal to 0: f(0) = 2(0) + 1 = 1.
Step 4: Next, we need to find the limit of f(x) as x approaches 0 from both sides.
Step 5: Calculate the left-hand limit (as x approaches 0 from the left): limit as x approaches 0- of f(x) = limit as x approaches 0- of x^2 = 0.
Step 6: Calculate the right-hand limit (as x approaches 0 from the right): limit as x approaches 0+ of f(x) = limit as x approaches 0+ of (2x + 1) = 1.
Step 7: Compare the left-hand limit, right-hand limit, and f(0): left-hand limit = 0, right-hand limit = 1, and f(0) = 1.
Step 8: Since the left-hand limit does not equal the right-hand limit, the function is not continuous at x = 0.

Piecewise Functions – Understanding how to evaluate and analyze functions defined by different expressions based on the input value.
Continuity – The concept of continuity at a point, which requires that the function value at that point equals the limit of the function as it approaches that point.
Limits – Calculating the limit of a function as it approaches a specific point from both sides.