Finding the range of a composite function? A really messed up one..?

Question

anonymous · Answer

$$f(x) = \frac{x^2-x+1}{x^2+x+1}$$Now I have to find f(f(x)). Is there an easier way than substituting? Because that's just gonna become ew. :(

anonymous · Answer

Oh I mean find the range of f(f(x)).

anonymous · Answer

the algebra to simplify is... ew.. but simplified it doesn't look too bad (thanks to mr. wolfram)

anonymous · Answer

Well, the thing is I don't get access to Mr. Wolfram during the exams :( (oh if only that would change...). So is that the only way?

anonymous · Answer

a simple way that is find conditions of x is belonging to real set.

anonymous · Answer

That's the range of f(x). I need the range of f(f(x)).

anonymous · Answer

@HairtUB
I hope the attachment is more like what you were looking for.

anonymous · Answer

Wow...you got it! Yeah I decided to plug in and it took me ages, but I got it and my answers are correct! :D Thank you so much for your time and help!