Question

Find the value of the constant k that makes the function continuous at x = 2.

f(x) = {
3x - k if x ≥ 2,
x² + 1 if x < 2
}

A. k = -2
B. k = -1
C. k = 1
D. k = 2

Answer 1

The value of the constant k that makes the function f(x) continuous at x = 2 is 1, as it satisfies the condition that the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.

Step-by-step explanation:

The student asked to find the value of the constant k that makes the function continuous at x = 2. For a function to be continuous at a certain point, the left-hand limit and the right-hand limit at that point must be equal to the function's value at that point.

In mathematical terms, we want to find k so that:

• lim (x→2-) f(x) = lim (x→2+) f(x) = f(2)

For x < 2, f(x) = x² + 1. So, the left-hand limit as x approaches 2 is:

(2)² + 1 = 4 + 1 = 5

For x ≥ 2, f(x) = 3x - k. So, the right-hand limit and the value of the function at x = 2 is:

3(2) - k = 6 - k

To make the function continuous, we set the limits equal to each other:

5 = 6 - k

Solving for k:

k = 6 - 5 = 1

Therefore, the constant k that makes the function continuous at x = 2 is 1.