There are three steps to solving a math problem.
- Figure out what the problem is asking.
- Solve the problem.
- Check the answer.
Example. Determine all values of x at which the function
is discontinuous.
Answer:
- Figure out what the problem is asking.
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:
- f(c) exists,
exists, and
For the function to be discontinuous at x = c, one of the three things above need to go wrong. Either
- f(c) is undefined,
doesn't exist, or
- f(c) and
both exist, but they disagree.
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.
- Solve the problem.
- Where is f(x) undefined?
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.
When the denominator is 0, either x = -3 or x = 4.
- Where does
not exist?
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.
- Where do f(c) and
both exist, but disagree?
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.
- Check the answer.
Besides doing the arithmetic again, probably the best thing to do is graph it with a 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.