Cancelling variable factors in a rational function

Question

Consider the function $\displaystyle\frac{2x−1}{x+5}$. The domain of this function is all real numbers except $x = -5$. Now consider that I do this: $\displaystyle\frac{2x−1}{x+5}⋅\frac{x}{x}$. This changes the domain of the function to all real numbers except $x = 5$ and $x = 0$. Why is this allowed? Technically x / x is just 1, but the two expressions end up having different domains just from multiplying by 1 = x / x. I just don't see why this is allowed considering that f(x) changed domains just from multiplying by that factor. In addition, why is the reverse situation allowed? For example, if we have a function defined by $\displaystyle f(x) =\frac{2x−1}{x+5}⋅\frac{x}{x}$, this means that the domain is all reals except -5 and 0. However, just by cancelling the x's, we have a whole new function, where x = 0 is now defined. However, this simple cancellation means that f(x) is no longer the same function, but a new function with a new domain, such that if $f\displaystyle (x)=\frac{x(2x−1)}{x(x+5)}$ then $\displaystyle f(x)≠\frac{2x−1}{x+5}$. I am just confused as to why this cancellation is allowed.