find the domain of the following rational function
r(x)= 2(x2-6x-40)
3(x2-100)
Unless otherwise specified, we usually take the domain of a function to be all real numbers for which the function is defined.
So consider what values of $\displaystyle x$ would make $\displaystyle r(x)$ undefined. Those are the values that are excluded from the domain. (Hint: look for values that would cause things like a division by zero or the taking of the square root/logarithm of a negative number).
Be careful. The domain is the values for which the function is defined. But when $\displaystyle x=\pm10$, the function is undefined (due to a division by zero). Thus your answer should be reversed: the domain is all real numbers not equal to $\displaystyle \pm10$. If you want to write this with set notation, there are several ways:
$\displaystyle \left\{x\in\mathbb{R}\;|\;x\ne10\text{ and }x\ne-10\right\}$
or
$\displaystyle (-\infty,\,-10)\cup(-10,\,10)\cup(10,\,\infty)$
for example.