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

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

Options:

Correct Answer: x = 1

Solution:

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

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

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

Step 1: Identify the point where we need to check continuity, which is x = 1.
Step 2: Calculate f(1) using the second part of the function since x = 1 falls into the category of x >= 1. f(1) = 2(1) - 1 = 1.
Step 3: Find the limit of f(x) as x approaches 1 from the left (x < 1). Use the first part of the function: limit as x approaches 1 from the left is f(x) = x^2. So, limit = 1^2 = 1.
Step 4: Find the limit of f(x) as x approaches 1 from the right (x >= 1). Use the second part of the function: limit as x approaches 1 from the right is f(x) = 2x - 1. So, limit = 2(1) - 1 = 1.
Step 5: Compare the values: f(1) = 1, limit from the left = 1, limit from the right = 1. Since all three values are equal, the function is continuous at x = 1.

Piecewise Functions – Understanding how to evaluate and analyze functions defined by different expressions based on the input value.
Continuity – Determining if a function is continuous at a point by checking the value of the function and the limits from both sides.
Limits – Calculating the left-hand and right-hand limits to assess the behavior of the function as it approaches a specific point.