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

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

Options:

Correct Answer: 1

Solution:

To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.

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

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

Step 1: Identify the point where we want to check continuity. In this case, it is x = 1.
Step 2: Find the limit of f(x) as x approaches 1 from the left (x < 1). This means we use the first part of the function: f(x) = x^2.
Step 3: Calculate the limit as x approaches 1 from the left: limit as x -> 1- of f(x) = limit as x -> 1- of x^2 = 1^2 = 1.
Step 4: Now, find the limit of f(x) as x approaches 1 from the right (x >= 1). This means we use the second part of the function: f(x) = 2x - 1.
Step 5: Calculate the limit as x approaches 1 from the right: limit as x -> 1+ of f(x) = limit as x -> 1+ of (2x - 1) = 2(1) - 1 = 1.
Step 6: Compare the two limits. Since both limits (from the left and right) equal 1, we conclude that f(x) 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 limits from both sides and the function's value at that point.
Limits – Calculating the left-hand and right-hand limits to assess continuity at a specific point.