Find numbers a and b or k so that f is continuous at every point. \
F(x)= 4x+7 if x<-4
kx+10 if x≥-4
Is -5/2 the correct answer?
4(-4) + 7 = k(-4) - 4
-4k = -5
k = +5/4
4(-4)+7 = k(-4) + 10
-19 = -4k
k = 19/4
To determine if f(x) is continuous at every point, we need to ensure that the limits from both sides of x = -4 are equal.
Let's first find the limit from the left side, as x approaches -4:
lim(x→-4-) f(x) = lim(x→-4-) (4x + 7)
Substituting -4 for x:
lim(x→-4-) (4(-4) + 7) = lim(x→-4-) (-16 + 7) = lim(x→-4-) -9 = -9
Now, let's find the limit from the right side, as x approaches -4:
lim(x→-4+) f(x) = lim(x→-4+) (kx + 10)
Substituting -4 for x:
lim(x→-4+) (k(-4) + 10) = lim(x→-4+) (-4k + 10) = -4k + 10
For f(x) to be continuous at x = -4, the two limits must be equal. Therefore, we need to solve the equation:
-9 = -4k + 10
Rearranging the equation:
-4k = -9 - 10
-4k = -19
Dividing both sides by -4:
k = -19/-4
k = 19/4
k = 4.75
Hence, the correct value for k is 4.75, not -5/2.
To determine if the function f(x) is continuous at every point, we need to check if the limit of f(x) as x approaches -4 from the left is equal to the value of f(x) at x = -4.
First, let's find the limit of f(x) as x approaches -4 from the left:
lim(x → -4-) (4x + 7)
To do this, we substitute -4 into the expression:
lim(x → -4-) (4(-4) + 7) = lim(x → -4-) (-16 + 7) = lim(x → -4-) (-9) = -9
Next, we need to find the value of f(x) at x = -4:
f(-4) = k(-4) + 10
Since f(x) is defined as kx + 10 when x is greater than or equal to -4, the value of f(-4) is equal to:
f(-4) = k(-4) + 10 = -4k + 10
For the function f(x) to be continuous at every point, the limit of f(x) as x approaches -4 from the left (-9) must be equal to the value of f(-4) (-4k + 10). Therefore, we set these two expressions equal to each other:
-9 = -4k + 10
Now we can solve for k:
-9 - 10 = -4k
-19 = -4k
Divide both sides by -4:
k = -19/-4
k = 19/4
So the correct value of k that makes the function continuous is k = 19/4, not -5/2.