<s>[INST] simplifying complex fractions when proving inverses with function composition? I'm working with two functions, $f(x)=\frac{x-3}{x+4}$ and $g(x)=\frac{4x+3}{1-x}$.
I need to simplify $f(g(x))$ and its opposite, but i'm not sure of the procedures regarding the more complicated forms like 
$$\frac{\frac{4x+3}{1-x}-3}{\frac{4x+3}{1-x}+4}$$
 [/INST] with the substitution $x \rightarrow g(x)$ the numerator of $f(x)$ becomes:
$$
x-3 \rightarrow \frac{4x+3}{1-x}-3=\frac{7x}{1-x}
$$
and the denominator becomes
$$
x+4 \rightarrow \frac{4x+3}{1-x}+4=\frac{7}{1-x}
$$
so
$$
f(g(x))= \frac{\frac{7x}{1-x}}{\frac{7}{1-x}}= x
$$
</s>