I have been informed that F(x)=X*((X+1)/(X+1)) is not equal to Y=X since the former is undefined at X=-1

So, F(x)=X ((X+1)/(X+1)) is not equal to Y=(1)X, am I then saying that (X+1)/(X+1) is not equal to (1)?

At this time to me this seems to imperil some pretty basic operations, for example take combining fractions.

D(x) = (1/x) + 2/(1+X) = 2

I have been taught that we combine fractions by multiplying each fraction by 1, in this case (X/X) and (X+1)/(X+1). Does this procedure increase the number of points at which the function D(x) is undefined? Or, wouldn't D(x) become some new function?

In addition, any function can be multiplied by one and remain unchanged. So P(x) can be said to have an infinite string of 1's being multiplied to it. If, (X+1)/(X+1) = 1, is true and F(x) is also undefined at -1 then couldn't it be said that a function U(x) = (X+2)/(X+2) exits and that F(x)*U(x) is equal to one and undefined at -2 and -1. Repeating this procedure for all, lets say counting numbers, would mean that any function is undefined for all counting numbers? This makes me think that no function can exist as it would be undefined at all possible numbers.

This whole thing seems clearly wrong to me.

I couldn't think of a better spot for this question within the forum structure. As far as I know this question of mine doesn't relate to any specific class, please feel free to move it as is appropriate.

Also I am interested in both narrow and broad corrections/comments. If this is more then just a simple error (as unlikely as that is) and relates to something complicated please let me know what that complicated thing is called.