Question

Let f(x) be defined as below. What value(s) of k would make the function continuous? f(x)= ⎩

⎨
⎧
​
x 2
+9x+18
x 2
−x−12
​
,
kx+2,
​
x<−3
x≥−3
​

Answer 1

To make the function continuous at the point x = -3, we need the following conditions to be satisfied:

1. The limits from the left and right must be equal.

2. The value of the function at x = -3 must also equal this limit.

Let's start by finding the limits from the left and right:

1. Limit from the left (x < -3):

lim(x -> -3-) f(x) = lim(x -> -3-) (x^2 + 9x + 18) = (-3)^2 + 9*(-3) + 18 = 9 - 27 + 18 = 0

2. Limit from the right (x >= -3):

lim(x -> -3+) f(x) = lim(x -> -3+) (kx + 2) = k*(-3) + 2 = -3k + 2

Now, for the function to be continuous at x = -3, these two limits must be equal, so we have:

0 = -3k + 2

Solving for k:

-3k = -2

k = 2/3

So, the value of k that would make the function continuous at x = -3 is k = 2/3.

Explanation: