If two functions are defined in terms of one another, can their composite be defined?

Question

Say $$f(x) = 3*g(x)$$ and $$g(x) = 3*f(x)$$ Is the function composition $f(g(x))$ defined?

Answer 1

Assume that for $x\in\mathbb{R} $, $f(x) $ and $g(x) $ are defined. Then $$ g(x) =\frac{f(x)} {3} \implies 3f(x) =\frac{f(x)}{3} $$ $$ 3f(x) =\frac{f(x)}{3} \implies 9f(x)=f(x) $$ $$ 9f(x)=f(x) \implies f(x) (9-1)=0 $$ Since $9-1=8\neq 0$, $f(x) =0$ which also implies $g(x) =0$. Then $f(g(x)) =f(0)=0$. In fact, this is true for all $$f(x) = ng(x) $$ $$g(x) = mf(x) $$ with $n, m \in \mathbb{R}$ and $nm \neq 1$. If $nm =1$ then $f(x)=g(x) $ is not defined so $f(g(x)) =f(f(x)) $ is not defined.