Find the value of m such that the function f(x) = { x^2 + m, x < 1; 4 - x, x

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Question: Find the value of m such that the function f(x) = { x^2 + m, x < 1; 4 - x, x >= 1 } is continuous at x = 1.

Options:

Correct Answer: 1

Solution:

Setting the two pieces equal at x = 1: 1^2 + m = 4 - 1. Solving gives m = 2.

Find the value of m such that the function f(x) = { x^2 + m, x < 1; 4 - x, x >= 1 } is continuous at x = 1.

Find the value of m such that the function f(x) = { x^2 + m, x < 1; 4 - x, x >= 1 } is continuous at x = 1.

Step 1: Identify the two pieces of the function f(x). The first piece is x^2 + m for x < 1, and the second piece is 4 - x for x >= 1.
Step 2: To find the value of m that makes the function continuous at x = 1, we need to set the two pieces equal to each other at x = 1.
Step 3: Substitute x = 1 into the first piece: 1^2 + m = 1 + m.
Step 4: Substitute x = 1 into the second piece: 4 - 1 = 3.
Step 5: Set the two results equal to each other: 1 + m = 3.
Step 6: Solve for m by subtracting 1 from both sides: m = 3 - 1.
Step 7: Simplify the equation: m = 2.

Piecewise Functions – Understanding how to evaluate and ensure continuity at a point where a piecewise function changes its definition.
Continuity – The concept that a function is continuous at a point if the limit from the left equals the limit from the right and equals the function value at that point.
Solving Equations – The ability to set equations equal to each other and solve for unknown variables.