Final Answer:
The constant of c that makes the function continuous is 4.
Explanation:
To find the value of the constant c that makes the function continuous, we need to ensure that the limit of the function as it approaches the point of discontinuity from both sides is equal. Let’s consider a piecewise function f(x) = { x² - 2x + c, if x < 3; 5x - 7, if x >= 3 }. To make this function continuous at x = 3, we need to find the value of c such that the limit of f(x) as x approaches 3 from the left is equal to the limit of f(x) as x approaches 3 from the right.
First, let’s find the limit of f(x) as x approaches 3 from the left. Substituting x = 3 into the first part of the function gives us (3)² - 2(3) + c = 9 - 6 + c = 3 + c. Next, let’s find the limit of f(x) as x approaches 3 from the right. Substituting x = 3 into the second part of the function gives us 5(3) - 7 = 15 - 7 = 8. To make the function continuous at x = 3, we need to set these two limits equal to each other: 3 + c = 8. Solving for c gives us c = 5 - 3, which simplifies to c = 4.
Therefore, the constant of c that makes the function continuous is indeed 4.