Question about combining functions.

Question

Is $F(1/g(x))=F(g(x))$ enough to say that $F(1/x)=F(x)$. Seems pretty reasonable but just want to make sure. Also keep in mind that I'm not worrying about domain and range of $g(x)$.

Answer 1

This would be true of the range of $g$ includes all non-zero real numbers; otherwise it could be false.

Answer 2

No! If e.g. $g$ is the constant function prescribed by $x\mapsto1$ then the premisse implies nothing.

From $F(1)=F(1)$ you cannot conclude that e.g. $F(\frac12 )=F(2)$.