Final answer:
The value of c that makes the function continuous is 27, as determined by setting the left-hand and right-hand expressions for f(x) equal at the point x = 3.
Step-by-step explanation:
To find the value of c that makes the function continuous, we consider the given piecewise function:
- f(x) = 2x + 9 for x ≤3
- f(x) = -4x + c for x > 3
A function is continuous at a point if the left-hand limit, the right-hand limit, and the value of the function at that point are all equal. Since we want the function to be continuous at x = 3, we set the expressions for f(x) from the left and right of x = 3 equal to each other:
2(3) + 9 = -4(3) + c
By calculating both sides, we find:
6 + 9 = -12 + c
This simplifies to:
15 = -12 + c
Therefore, adding 12 to both sides gives us:
c = 27
So, c = 27 is the value that ensures the function is continuous at x = 3.