For which value of a is the function f(x) = { x^2 + a, x < 1; 3, x >= 1 }

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Question: For which value of a is the function f(x) = { x^2 + a, x < 1; 3, x >= 1 } continuous at x = 1?

Options:

Correct Answer: 2

Solution:

To ensure continuity at x = 1, we set limit as x approaches 1 from left (1 + a) equal to f(1) = 3, thus a = 2.

For which value of a is the function f(x) = { x^2 + a, x < 1; 3, x >= 1 } continuous at x = 1?

For which value of a is the function f(x) = { x^2 + a, x < 1; 3, x >= 1 } continuous at x = 1?

Step 1: Identify the function f(x) which is defined in two parts: f(x) = x^2 + a for x < 1 and f(x) = 3 for x >= 1.
Step 2: To check for continuity at x = 1, we need to 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 + a.
Step 3: Calculate the limit as x approaches 1 from the left: limit as x approaches 1 of (x^2 + a) = 1^2 + a = 1 + a.
Step 4: Now, find the value of f(1) using the second part of the function since x = 1 falls in that range: f(1) = 3.
Step 5: For the function to be continuous at x = 1, the limit from the left (1 + a) must equal f(1) (which is 3). So, we set up the equation: 1 + a = 3.
Step 6: Solve for a by subtracting 1 from both sides: a = 3 - 1 = 2.

Continuity of Functions – Understanding the conditions under which a piecewise function is continuous at a given point.
Limits – Applying the concept of limits to evaluate the behavior of a function as it approaches a specific point from different directions.
Piecewise Functions – Analyzing functions defined by different expressions based on the input value.