TITLE: How can I convince my math teacher?
QUESTION [16 upvotes]: Ok, so I got an answer wrong on my exam because my teacher says that the function $f(x)=\frac{(x+2)x}{x+2}=x$ but I insist that it isn't defined for x=-2. If it was then $\frac{x}{x}=1$ for all reals and so $\frac{0}{0}=1$. However this doesn't seem to do the trick with my teacher. How can I use the latter fact to prove something outrageous and convince her?

REPLY [4 votes]: By convention, we assume that the domain of such functions is the larges subset of $\mathbb{R}$ for which the function is defined. So, the domain of $f(x)=x\dfrac{(x+2)}{(x+2)}$ is $\mathbb{R}\setminus\{-2\}$. Additionally, for all $x\ne -2$, we can simplify the function to $f(x)=x$. But we still have the domain of this function as $\mathbb{R}\setminus \{-2\}$. You may try asking your teacher whether they think that $f(x)=\dfrac{x+2}{x+2}$ is defined at $x=-2$ or not.
Another thing that might help:
Ask them whether they think the function $f(x)=\dfrac{x^5-x^2+3x-3}{x^3+2x^2-3}$ is defined at $x=1$. If they say no, then tell them that the function is equivalent to $\dfrac{(x^4+x^3+x^2+3)(x-1)}{(x^2+3x+3)(x-1)}$ and so from their line of reasoning, the function should equal $\dfrac67$ at $x=1$.