Question 115062
{{{f(x)=(x^2-1)/(sqrt(x+1)+sqrt(x+1))}}}


This function is undefined if {{{x+1<=0}}}, or putting a positive spin on the statement, the function is defined if and only if {{{x+1>0}}} => {{{x>-1}}}.  Therefore, the domain of the function is the interval (-1,{{{infinity}}}]


The zeros of the function can be seen readily if you factor the numerator and then rationalize the denominator.


{{{f(x)=((x+1)(x-1))/2sqrt(x+1)}}}
{{{f(x)=((x+1)(x-1)sqrt(x+1))/2sqrt(x+1)sqrt(x+1)}}}
{{{f(x)=((x+1)(x-1)sqrt(x+1))/(2x+2)}}}


{{{f(x)=0}}} iff {{{x-1=0}}} or {{{x+1=0}}} or {{{sqrt(x+1)=0}}}


{{{x-1=0}}} iff {{{x=1}}}, so {{{f(1) = 0}}} and 1 is a zero of the function.
{{{x+1=0}}} iff {{{x=-1}}}, but -1 is not in the domain of the function, therefore {{{f(-1)}}} is undefined (notice the open ended domain interval) and -1 is not a zero of the function.
{{{sqrt(x+1)=0}}} iff {{{x+1=0}}} => {{{x=-1}}}, and again, -1 is not in the domain.


Therefore the only zero of the function is 1.


{{{graph(600,600,-10,10,-10,10,(x^2-1)/(sqrt(x+1)+sqrt(x+1)))}}}


As to the balance of your question, "determine the signs of f where..."  Where what?  Re-post with the complete question and someone will try to answer.