# Sketch the graph of F. F(x) = (x^2-1)/|x-1| ???

### 1 Answer | Add Yours

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.

So:

when x>1, |x-1|=x-1

Note that `x^2-1=(x-1)(x+1)`

So: if x>1, then:

`F(x)=((x-1)(x+1))/(x-1)=x+1`

The graph of the function of f(x)=x+1 is:

Now, if x<1, then |x-1|=-(x-1)

So

`F(x)=((x-1)(x+1))/(-(x-1))=(x+1)/(-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)