Final answer:
To make the piecewise function continuous at x = 3, we solve for c by setting c² - 9 equal to 3 + c, resulting in the quadratic equation c² - c - 12 = 0. Using the quadratic formula, we find the values of c are 4 and -3.
Step-by-step explanation:
The question asks us to find the values of c so that the piecewise function f(x) is continuous. A piecewise function is continuous at a point if both the left-hand limit and the right-hand limit as x approaches the point from either side are equal to the function's value at that point. In this case, we need the left-hand limit of c² - x² as x approaches 3 to equal the right-hand limit of x + c as x approaches 3.
Setting the two expressions equal to each other when x is equal to 3 gives us:
c² - (3)² = 3 + c
c² - 9 = 3 + c
c² - c - 12 = 0
This is a quadratic equation, and solving for c using the quadratic formula yields:
c = (-(-1) ± √((-1)^2 - 4(1)(-12)))/(2(1))
c = (1 ± √(1 + 48))/2
c = (1 ± √49)/2
c = (1 ± 7)/2
Therefore, the two possible values of c are:
- c = (1 + 7)/2 = 4
- c = (1 - 7)/2 = -3