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

Jan 23, 2015

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 $f \left(x\right) = \setminus \frac{1}{x - 2}$, 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 $f \left(2\right) = \setminus \frac{1}{2 - 2}$, 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:
${x}^{2} - 49 = 0 \setminus \iff {x}^{2} = 49 \setminus \iff x = \setminus \pm \setminus \sqrt{49} \setminus \iff x = \setminus \pm 7$.

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

With a proper notation, your domain is the set $\left\{x \setminus \in \setminus m a t h \boldsymbol{R} | x \setminus \ne \setminus \pm 7\right\}$