Question

What is the domain of the function `f(x)=(3x^2)/(x^2-49)`?

Answer 1

The domain of a function is the set of the values in which you can calculate the function itself.

The problem of finding the domain of a functions is due to the fact that not every function "accepts" every real number as an input.

If you have a rational function, you must exclude from the domain the values for which the denominator is zero, since you cannot divide by zero. So, for example, if you consider the function `f(x)=\frac{1}{x-2}`, you see that you can evaluate it for every real value `x`, as long as it is not 2: in that case, you would have `f(2)=\frac{1}{2-2}`, an obviously invald operation. So, we say that the domain of `f` is the whole real number set, deprived of the element "2".

In your case, the denominator is the function `x^2-49`. Let's see for which values it is zero:
`x^2-49=0 \iff x^2=49 \iff x=\pm\sqrt{49} \iff x=\pm 7`.

These are the only two values you have to consider, since for any other `x\ne \pm 7`, the denominator isn't zero, and you have no problem calculating `f(x)`.

With a proper notation, your domain is the set `{x \in \mathbb{R} | x\ne\pm7}`