Reductions of functions

I don't know TeX,so please bare with me.

Say we have two functions:

f(x)=x²-2x+1 / x-1
g(x)=x-1

We can reduce the function f(x) to x-1.So,we could say that f(x)=g(x).

One problem-the function f(x) is not defined at x=1,but g(x) is.

Questions:Why can we reduce the function f(x) to x-1,but it is not defined at x=1??

And in general-wtf,/sci/?

>>2
But if x=1 in g(x),then g(1)=0.
Maybe I didn't explain the question properly:
If f(x) can be reduced to g(x),then why is it that f(x) is defined at 1,and g(x) is not?