The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous a

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Question: The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?

Options:

Correct Answer: 1

Solution:

To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.

The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?

The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?

Step 1: Identify the function f(x) and the point we want to check for continuity, which is x = 1.
Step 2: Find the left limit as x approaches 1. This means we look at the part of the function where x < 1, which is f(x) = x + 2.
Step 3: Calculate the left limit: f(1) = 1 + 2 = 3.
Step 4: Find the right limit as x approaches 1. This means we look at the part of the function where x > 1, which is f(x) = x^2.
Step 5: Calculate the right limit: f(1) = 1^2 = 1.
Step 6: Check the value of the function at x = 1. The function is defined as f(1) = 3.
Step 7: Compare the left limit (3) and the right limit (1) with the value of the function at x = 1 (3).
Step 8: Since the left limit (3) does not equal the right limit (1), f(x) is not continuous at x = 1.

Piecewise Functions – Understanding how piecewise functions are defined and how to evaluate them at specific points.
Continuity – The definition of continuity at a point, which requires that the left limit, right limit, and function value at that point are all equal.
Limits – Calculating left-hand and right-hand limits to determine continuity.