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

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

Options:

Correct Answer: Not continuous at x = 1

Solution:

The function is not continuous at x = 1 because the left-hand limit does not equal the function value.

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

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

Step 1: Understand the function f(x). It has three parts: x^2 when x is less than 1, 2 when x is exactly 1, and x + 1 when x is greater than 1.
Step 2: Find the left-hand limit as x approaches 1. This means we look at values of x that are just less than 1. For these values, f(x) = x^2. So, we calculate the limit: limit as x approaches 1 from the left is 1^2 = 1.
Step 3: Find the value of the function at x = 1. According to the function definition, f(1) = 2.
Step 4: Compare the left-hand limit and the function value at x = 1. The left-hand limit is 1, and the function value is 2. Since 1 does not equal 2, they are not the same.
Step 5: Conclude that the function is not continuous at x = 1 because the left-hand limit does not equal the function value.

Piecewise Functions – Understanding how piecewise functions are defined and how to evaluate them at specific points.
Continuity – Determining the continuity of a function at a point by checking limits and function values.
Limits – Calculating left-hand and right-hand limits to assess continuity.