If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f(x) continuous at x

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Question: If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f(x) continuous at x = 1? (2019)

Options:

Correct Answer: Yes

Exam Year: 2019

Solution:

At x = 1, f(1) = 1^2 = 1 and the limit from the left is also 1, hence f(x) is continuous at x = 1.

If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f(x) continuous at x = 1? (2019)

If f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1, is f(x) continuous at x = 1? (2019)

Step 1: Identify the function f(x). It is defined in two parts: f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1.
Step 2: Find the value of f(1). Since 1 is greater than or equal to 1, we use the second part of the function: f(1) = 2(1) - 1 = 1.
Step 3: Calculate the limit of f(x) as x approaches 1 from the left (x < 1). We use the first part of the function: limit as x approaches 1 from the left is f(x) = x^2 = 1^2 = 1.
Step 4: Calculate the limit of f(x) as x approaches 1 from the right (x ≥ 1). We use the second part of the function: limit as x approaches 1 from the right is f(x) = 2(1) - 1 = 1.
Step 5: Compare the value of f(1) with the limits from both sides. Since f(1) = 1, limit from the left = 1, and limit from the right = 1, all are equal.
Step 6: Conclude that f(x) is continuous at x = 1 because the value of the function at that point equals the limits from both sides.

Piecewise Functions – Understanding how piecewise functions behave at transition points.
Continuity – Determining if a function is continuous at a specific point by checking limits and function values.
Limits – Calculating left-hand and right-hand limits to assess continuity.