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

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

Options:

Correct Answer: Not continuous

Solution:

The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.

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

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

Step 1: Identify the function f(x) which is defined in three parts: f(x) = x^2 for x < 0, f(x) = 0 for x = 0, and f(x) = x + 1 for x > 0.
Step 2: Find the left limit as x approaches 0. This means we look at values of x that are slightly less than 0. For these values, f(x) = x^2. As x gets closer to 0 from the left, x^2 approaches 0.
Step 3: Calculate the left limit: lim (x -> 0-) f(x) = 0.
Step 4: Find the right limit as x approaches 0. This means we look at values of x that are slightly greater than 0. For these values, f(x) = x + 1. As x gets closer to 0 from the right, x + 1 approaches 1.
Step 5: Calculate the right limit: lim (x -> 0+) f(x) = 1.
Step 6: Compare the left limit and the right limit. The left limit is 0 and the right limit is 1.
Step 7: Since the left limit (0) is not equal to the right limit (1), the function is not continuous at x = 0.

Piecewise Functions – Understanding how piecewise functions are defined and how to evaluate them at different intervals.
Limits and Continuity – Analyzing the left-hand and right-hand limits to determine the continuity of a function at a specific point.