On an exam we were asked: what is the domain of the following function?
$$\large f(x)=\frac{\frac{x}{x-2}}{\frac{2x+7}{x-1}}$$
My solution:
This is the same as $$\large \frac{\frac{x}{x-2}}{1}\times\frac{1}{\frac{2x+7}{x-1}}$$ so the domain is $\mathbb R\backslash\{1,2,-\frac72\}$
Teacher's solution:
Use sandwich law to see it is the same as $\frac{x(x-1)}{(2x+7)(x-2)}$ and therefore the domain is $\mathbb R\backslash\{-\frac72,2\} $
I argue you need to look at the order of operations and see first you must evaluate the two halfs independently, but I can't convince her.
Your answer is correct; her answer is for the function without its removable singularity at $x=1$, which has a different domain. That is like arguing the domain of $(x+1)/(x^2-1)$ only excludes $x=-1$ because it can be reduced to $1/(x-1)$ -- while they're identical almost everywhere their domains differ because of the point $x=-1$.