given f(x)= { 3 - x^2, x<2 and kx - 5, x ≥ 2 what value of k would make f(x) continuous at x=2?

Question

I understand what this question is asking but I don't know what steps to do.

Also, I don't get how to do the next part of the question: state the ordered pair at the transition point where the function becomes continuous.

Is it possible to do that without graphing it?

Answer 1

The function f(x) is a piecewise function, meaning it's sort of like two different functions stitched together. Each piece of the function doesn't necessarily need to end where the next one begins; if that is the case, then it is not continuous.

In this problem, the functions are stitched together at x = 2 and what you're trying to do is find a value for k such that the first function (3 - x²) does end where the next one (kx - 5) begins. How can we find such a k? Well if we want our piecewise function to be continuous, then that means our value at x = 2 should be the same when approached from the first and second functions. So we can find k where both functions produce the same value at x = 2 by setting the functions equal to each other, plugging in 2 for x, and then solving for k. Hopefully, with that explained, you can try to do that part yourself, but if you need help, feel free to comment for further clarification.

Once you know what k is, you can solve the next part of the question, which is simply asking what that point (x, y) at x = 2 is. That's just a matter of plugging in 2 for either one of the functions (or both! Make sure you get the same value) and evaluating. Good luck!