Sketch the graph of F. F(x) = (x^2-1)/|x-1| ???
If x>1, then x-1 > 0, so the absolute value doesn't change anything
If x<1, then x-1 < 0, so the absolute value takes a negative quantity and turns it positive (in other words, the absolute value is the same as multiplying by -1)
If x=1, then x-1=0, so F has division by 0, and is undefined.
when x>1, |x-1|=x-1
Note that `x^2-1=(x-1)(x+1)`
So: if x>1, then:
The graph of the function of f(x)=x+1 is:
Now, if x<1, then |x-1|=-(x-1)
The graph of g(x)=-x-1 is:
So the graph of F looks like f(x), or the first graph, if x>1
and it looks like g(x), or the second graph, if x<1
(and it is undefined when x=1)
so we want to splice these two pictures together at x=1:
This is the graph of F(x)