f is not equal to g. When you factorize f(x) and cancel out some terms, it does give f(x)=x, which is the same function as g(x). Even though they seem equal, they are not equal because f(x)=x is not the original function we are talking about here. The original function has no answer at x=1 because it gives zero divided by zero. Meanwhile, g(1) is equal to 1. Therefore, f is not equal to g.
Work Step by Step
f(x)=(x²-x)/x-1 f(x)=x(x-1)/x-1 f(x)=x g(x)=x f(x)=(x²-x)/x-1=(1²-1)/1-1=0/0=undefined g(x)=x=1