Question: I understand that division by zero isn't allowed, but we merely just multiplied $f(x) = 1$ by $\frac{x-1}{x-1}$ to get $f(x) = \frac{x-1}{x-1}$ and $a\cdot 1 = 1\cdot a = a$ so they're the **same function** but with **different domain** how is this possible?

*Or in other words* why don't we simplify $f(x) = \frac{x-1}{x-1}$ to $f(x) = 1$ before plotting the points.
Is it just defined this way or is there a particular reason ?

**Note:** my book says the domain of $f(x) = 1$ is $\mathbb{R}$ and the domain of 
$f(x) = \frac{x-1}{x-1}$ is $\mathbb{R}$ except $1$.

Answer: 
**Question**: What is a function? 

**Answer**: Maybe simply said it is a map (receipe), $f(x)$, that projects some elements, $x$, contained in a specifically defined set, Domain $D$, into another set, Range $R$.

**Discussion**: Hence when defining a function one must define the Domain as well as the functional form. Otherwise the function is not defined.

**Conclusion**: If two funtions have the same domain and the same receipe then they are the same "maps" otherwise they are not.
$f(x)=(x-1)/(x-1)$ does not have a value when $x=1$, different thing happens to $f(x)=1$