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

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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:

The left limit as x approaches 1 is 1, and the right limit is also 1. Thus, f(1) = 1, making it continuous.

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 left limit as x approaches 1. This means we will use the part of the function for x < 1, which is f(x) = x^2.
Step 3: Calculate the left limit: lim (x -> 1-) f(x) = lim (x -> 1-) x^2 = 1^2 = 1.
Step 4: Find the right limit as x approaches 1. This means we will use the part of the function for x >= 1, which is f(x) = 2x - 1.
Step 5: Calculate the right limit: lim (x -> 1+) f(x) = lim (x -> 1+) (2x - 1) = 2(1) - 1 = 1.
Step 6: Check the value of the function at x = 1. Since x = 1 falls in the second part of the function, f(1) = 2(1) - 1 = 1.
Step 7: Compare the left limit, right limit, and the value of the function at x = 1. All three are equal to 1.
Step 8: Since the left limit, right limit, and the function value at x = 1 are all equal, the function is continuous at x = 1.

Continuity of Functions – Understanding how to determine if a piecewise function is continuous at a specific point by evaluating limits and function values.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value.
Limits – Calculating left-hand and right-hand limits to assess continuity at a point.