after we factor the denominator, we can cancel x-1 in the numerator. So the function should be defined for x=1 right?
No, cancelling terms work for numbers but not for algebraic expressions. It is like saying 0/0 equals to 1 coz it gets cancelled in both the numerator and denominator.
For the question you have mentioned, imagine you want to find the value at x=1. If you cancel the terms out, you’ll end up with a valid answer, but infact f(1) is not defined.
PS: Same goes for cross multiplication of algebraic expressions. Cross multiplication will remove all the undefined values from the original answer as there will not be any denominator left after cross multiplying, so be careful!