Example:
$f(x) = \frac{x^2+2x-15}{x-3}$
Can be factored and rewritten as:
$f(x) = (x+5)$
At $x = 3$, the top, unaltered function yields undefined, while the bottom yields 8.
If I am correct in my understanding, a factored version of an expression is essentially a rewritten but still equivalent form of the original expression. No matter the form of the expression or equation, I was under the impression that all forms of the same expression would maintain equivalency. Is it that functions can sometimes have different outcomes but still maintain equivalency as long as the only difference is in an undefined value or values?
Is there a strict definition on "equivalency"? and is there a rule or law or anything concerning this?
Thank you for your time and help.
There is!
As functions, they have different expressions and exist at different points, so they are NOT equivalent of course.
However, it is true that: $$\lim_{x\to c}\dfrac{x^2+2x-15}{x-3}=\lim_{x\to c}x+5$$ (for any $c$)
So since the limits hold up on both sides, you'd could remove the discontinuity of the first expression by replacing it by the second.
However, Matti P.'s comment is also correct - you need to explicitly state that the function does not exist at $x=-3$, because it doesn't without changing it. The act of "removing the discontinuity" is itself changing the function.