Help reading functions with denominator

I've been having issues seeing functions that have denominators. I have few issues understanding most regular functions and how they are manipulated but in my effort to return to school I have missed out on understanding functions when combined with a denominator.

For example:

Is there any method to seeing that the lines are at +1/2 and -1/2? Is there a good website that can teach me methods of visualizing various fraction functions?

Re: Help reading functions with denominator

Maybe rewrite it as then it's easier to see that f(x)=1/2 when 1-x>0 and f(x)=-1/2 when 1-x<0.

I'd also recommend f(x)=|1-x|/(2(1-x)) - Wolfram|Alpha and you could download a graph plotter to play with.