Find the value of constants c and d that make the function below continuous at
x = 4.
f(x) =
x2 − 3x x < 4
c x = 4
d + x x > 4
c =
d = 8
I got d=8, but finding c is giving me trouble.
may be if you put your parentheses well i might be able to help you,because i dont understand you mathematical write up
To find the value of constant c that makes the function continuous at x = 4, we need to equate the limits of the function from the left and right sides as x approaches 4.
From the left side (x < 4), the limit of the function is:
lim(x→4-) (x^2 - 3x)
To find this limit, substitute x = 4 into the function:
(4^2 - 3(4)) = (16 - 12) = 4
From the right side (x > 4), the limit of the function is:
lim(x→4+) (d + x)
To find this limit, substitute x = 4 into the function:
(d + 4)
For the function to be continuous at x = 4, the left and right-sided limits must be equal. Therefore, we set the two limits equal to each other:
4 = (d + 4)
Now we can solve for d by subtracting 4 from both sides:
4 - 4 = d + 4 - 4
0 = d
Therefore, d = 0.
To find the value of constant c, we know that the function is equal to c at x = 4:
c = (4^2 - 3(4))
c = (16 - 12)
c = 4
So, the values of constants c and d that make the function continuous at x = 4 are:
c = 4
d = 0
To find the value of constant c that makes the function continuous at x = 4, we need to ensure that the left-hand limit and the right-hand limit of the function are equal at x = 4.
The left-hand limit is found by evaluating the function as x approaches 4 from the left side. In this case, x < 4. So substituting x = 4 into the first part of the piecewise function:
f(x) = x^2 - 3x
f(4) = 4^2 - 3(4) = 16 - 12 = 4
The right-hand limit is found by evaluating the function as x approaches 4 from the right side. In this case, x > 4. So substituting x = 4 into the third part of the piecewise function:
f(x) = d + x
f(4) = d + 4
For the function to be continuous at x = 4, the left-hand limit and the right-hand limit must be equal, which means:
4 = d + 4
Solving for d, we get:
d = 4 - 4 = 0
Therefore, the value of constant d that makes the function continuous at x = 4 is d = 0.
As for constant c, since the first part of the piecewise function is c when x = 4, we haven't been given any specific conditions or constraints for c. Without further information, we cannot determine the exact value of c.
So, we have found d = 0, but we cannot determine the value of c without more information.