When considering a function how do you find an interval of monotonicity containing a certain value of $x$? The question is part of an online quiz and gives me a function and the a value of $x$. I am then asked to find the upper and lower bounds of the monotonic interval that this value of $x$ is contained within. Honestly I'm not even sure where to start I haven't been taught this. I'm not able to provide an example as the quiz was timed and I can't go back and look. The only thing I am familiar with is testing for monotonicity.
Thanks.

If your function was differentiable as mentioned in your comment, then it's increasing on the intervals that its derivative is positive and decreasing on the intervals that its derivative is negative. This is known as the first derivative criteria (at least at my country!).
Let's say you're given the function 
$$f(x) = x^2 - 3x + 2$$ 
as a simple example, then its derivative will be : 
$$f'(x) = 2x - 3$$
Then, it's obvious that it will be : 
$$f'(x) > 0 \Rightarrow x>3/2$$
$$f'(x) < 0 \Rightarrow x<3/2$$
This means that your function will be decreasing in the interval $(-\infty,3/2)$ and increasing in the interval $(3/2,\infty)$ with a stationary point at $x=3/2$ since there is $f'(x) = 0$.
The first derivative criteria holds for any differentiable function $f(x)$, not only for polynomials of course and in this way you can determine the intervals of monotonicity as mentioned in your question.
