There are three steps to solving a math problem.
Example. Determine all values of x at which the function
We want to make sure we understand the problem. What does discontinuous mean? It means "not continuous", but what does that mean?
A function is continuous at a value x = c if three things happen:
For the function to be discontinuous at x = c, one of the three things above need to go wrong. Either
This problem is asking us to examine the function f and find any places where one (or more) of the things we need for continuity go wrong.
Since we're looking at a rational function, f is undefined wherever its denominator is 0. To find where that is, we need to factor the numerator and denominator.
The denominator is 0, therefore f is undefined, at x = -3 and x = 4.
This rational function has a hole at x = -3 and a vertical asymptote at x = 4, therefore doesn't exist. This gives us another reason that f(x) is discontinuous at x = 4.
This function doesn't have any places like that! Since a rational function is continuous everywhere it's defined, we've found all the discontinuous places we need to worry about.
To summarize, this function is only discontinuous at x = -3 and x = 4.
Besides doing the arithmetic again, probably the best thing to do is graph it on the calculator. Make sure it looks continuous except at x = 4, where there should be an asymptote. If we ask the calculator what the function is for x = -3, it should say "ERROR," because f(-3) is undefined!