Because f(x) is odd function gx and odd function f[gx] is an even function.

So f(-x)=-f(x) g(-x)=-g(x)

f[g(-x)]=f[-g(x)]=-f[g(x)]

So f[gx] is odd function,

It is wrong that fx is odd function gx or odd function f[gx] is an even function.

I hope I can help you and adopt it. I wish you progress in your study.