Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 }

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Question: Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } at x = 1.

Options:

Correct Answer: Continuous

Solution:

At x = 1, f(1) = 2(1) - 1 = 1 and lim x→1- f(x) = 1, lim x→1+ f(x) = 1. Thus, f(x) is continuous at x = 1.

Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } at x = 1.

Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } at x = 1.

Step 1: Identify the function f(x) which 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 the function at x = 1. Since x = 1 falls in the second part of the function, calculate f(1) = 2(1) - 1.
Step 3: Calculate f(1). This gives us f(1) = 2 - 1 = 1.
Step 4: Find the left-hand limit as x approaches 1 (denoted as lim x→1- f(x)). Since x < 1, use the first part of the function: lim x→1- f(x) = lim x→1- x^2 = 1^2 = 1.
Step 5: Find the right-hand limit as x approaches 1 (denoted as lim x→1+ f(x)). Since x ≥ 1, use the second part of the function: lim x→1+ f(x) = lim x→1+ (2x - 1) = 2(1) - 1 = 1.
Step 6: Compare the value of the function at x = 1, the left-hand limit, and the right-hand limit. We found f(1) = 1, lim x→1- f(x) = 1, and lim x→1+ f(x) = 1.
Step 7: Since f(1) equals both the left-hand limit and the right-hand limit, conclude that f(x) is continuous at x = 1.

Piecewise Functions – Understanding how to evaluate and analyze functions defined by different expressions over different intervals.
Limits – Calculating the left-hand and right-hand limits to determine continuity at a specific point.
Continuity – Determining if a function is continuous at a point by checking if the limit equals the function value.