Question

At what point is the function y=(x+3)/(x^(2)-7x+12) continuous

Answer 1

If a function is defined as

$h(x)=\dfrac{f(x)}{g(x)}$

where both $f(x),g(x)$ are continuous functions, then $h(x)$ is also continuous where defined, i.e. where $g(x)\neq0$

So, in your case, this function is continous everywhere, except where

$x^2-7x+12=0$

To solve this equation, we can use the formula $x^2-sx+p=0$

It means that, if the leading terms is 1, then the x coefficient is the opposite of the sum of the roots, and the constant term is the product of the roots.

So, we're looking for two terms whose sum is 7, and whose product is 12. These numbers are easily found to be 3 and 4.

So, this function is continuous for every real number different than 3 or 4.