simplifying polynomial rational expressions

Question

When rational expression $$\frac{x^{2} + 3x}{x^{2} + 5x}$$ is simplified it equals $(x+3)/(x+5)$ where $x\ne 0$. What I don't understand is why are the two functions not equal for the whole domain while we are just doing is dividing for example $2/4 = 1/2$, in this case, we only divided. (I know this is a stupid question)

Answer 1

If we define the domain of a rational expression as the largest subset of $\mathbb R$ (or of $\mathbb C$ or whatever) at which that expression is defined, then the domain of $\dfrac{x^2+3x}{x^2+5x}$ is $\mathbb R\setminus\{0,-5\}$, whereas the domain of $\dfrac{x+3}{x+5}$ is just $\mathbb R\setminus\{-5\}$. Since they have distinct domains, they are not the same function.

Answer 2

The expressions are equivalent in the algebraic sense, but the functions are different.

For example, consider the functions $f(x)=\frac{x}{x}$ and $g(x)=1.$ The domains are $\operatorname{dom} f(x)=\mathbb{R}\setminus \{0\}$, and $\operatorname{dom} g(x)=\mathbb{R}.$ Thus, $f(x)\neq g(x)$, even though $\frac{x}{x}=1$ when $x\neq0$.