I have a function which looks like:
$$\frac{x+10}{(x+10) (x - 9) (x - 5)}$$
The domain of the function for input x is any number except -10, 9, and 5, because it would be dividing by zero. The thing I don't quite get is that if I were to cancel out the (x+10)'s
$$\frac{1}{(x-9)(x-5)}$$
I'm told the function has changed. I know that now the domain includes -10, when before it didn't, but this seems a bit paradoxical to me because
If I were to do:
$$\frac{x+10}{(x+10) (x - 9) (x - 5)} = a$$
then
$$\frac{1}{(x - 9) (x - 5)} also = a$$
Is there a way a someone could explain this to me because it seems a bit paradoxical. The previous two equations are equal, but the functions are not?
Is it the same function but just with a different domain or is it a completely different function?
In your reasoning with
$$ \frac{x+10}{(x+10)(x-9)(x-5)} = a $$
you assumed that $a$ is a real number. This happens when $x\neq -10$. Therefore, you inadvertently assumed that $x \neq -10$, in which case the two expressions are the same.
There is no paradox. The two expressions are the same if $x\neq -10$. If $x=-10$, you can easily see that the first expression is not defined, whereas the second expression does get a real value.