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

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

Options:

Correct Answer: 1

Solution:

To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.

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

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

Step 1: Identify the function f(x) which has two parts: 2x + 3 for x < 1 and x^2 + 1 for x >= 1.
Step 2: To check if f(x) is continuous at x = 1, we need to find the left limit as x approaches 1 from the left (x < 1).
Step 3: Calculate the left limit: f(1) when using the first part of the function (2x + 3). So, f(1) = 2(1) + 3 = 5.
Step 4: Now, find the right limit as x approaches 1 from the right (x >= 1).
Step 5: Calculate the right limit: f(1) when using the second part of the function (x^2 + 1). So, f(1) = (1)^2 + 1 = 2.
Step 6: Compare the left limit (5) and the right limit (2). Since they are not equal, f(x) is not continuous at x = 1.

Piecewise Functions – Understanding how piecewise functions are defined and how to evaluate them at specific points.
Limits – Calculating left-hand and right-hand limits to determine continuity at a point.
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.