I'm studying A-Level maths and a question that's come up in my textbook for Core 3 relates to composite functions. I believe I've got the correct answer but in the textbook it looks as if I haven't simplified the answer sufficiently. The question was as follows:
If $f(x)=x/(1-x)$ find $f(f(x))$
My answer was $$\frac{\frac{x}{1-x}}{1-\frac{x}{1-x}}$$ The listed solution simplifies the answer I got, as follows: $$\frac{\frac{x}{1-x}}{1-\frac{x}{1-x}}= \frac{x}{(1-x)-x}=\frac{x}{1-2x}$$
Apologies if this is basic stuff, but I don't quite follow how they've gone from $$\frac{\frac{x}{1-x}}{1-\frac{x}{1-x}}$$ to $$\frac{x}{(1-x)-x}$$ Would someone be able to explain?
It's $$\frac{\frac{x}{1-x}}{1-\frac{x}{1-x}}=\frac{\frac{x}{1-x}\cdot(1-x)}{\left(1-\frac{x}{1-x}\right)(1-x)}=\frac{x}{1\cdot(1-x)-\frac{x}{1-x}\cdot(1-x)}=\frac{x}{1-x-x}=\frac{x}{1-2x}$$