Question

Find the value of the constant b that makes the following function continuous on (−[infinity],[infinity])

. f(x)=4x−6 if x≤2

−2x+b if x>2

Answer 1

The value of the constant b that makes the function f(x) continuous on (−∞,∞) is 6. This ensures that the left-hand limit, right-hand limit, and the value of f(x) at x=2 are all equal.

Step-by-step explanation:

To find the value of the constant b that makes the function f(x) continuous on (−∞,∞), we need to ensure that the left-hand limit at x=2 equals the right-hand limit at x=2 as well as the value of f(x) at x=2 for both pieces of the function.

For x≤2, the function is defined as f(x)=4x−6, so f(2)=4(2)−6=2.

For x>2, the function is defined as f(x)=-2x+b. To be continuous at x=2, f(2) must also equal 2. So -2(2)+b=2 leading to the equation b=2+4=b=6.

For x ≤ 2, the function is f(x) = 4x - 6. Taking the limit as x approaches 2 from the left side, we have:

lim(x→2-) [4x - 6] = 4(2) - 6 = 2

For x > 2, the function is f(x) = -2x + b. Taking the limit as x approaches 2 from the right side, we have:

lim(x→2+) [-2x + b] = -2(2) + b = b - 4

Since the two limits must be equal for the function to be continuous at x = 2, we have:

2 = b - 4

Therefore, the value of b that makes the function continuous on (−∞,∞) is 6.