Is $f(g)$ homogeneous? If so, of what degree?

Question

Given $f$ and $g$ are homogeneous functions of degree $k$. I have to show if $f(g)$ is homogeneous or not, and if so, of what degree.

Definition (Homogeneous function). Let $f:\mathbb{R}^{n}\rightarrow \mathbb{R}$. We say that $f$ is homogeneous of degree $k$ if for all $t \in \mathbb{R}_{+}$ we have $$f(tx)=t^{k}f(x)$$ for all $x$.

So $$f(g(tx))= t^{k} f(g(tx))$$$$=t^{k}f(t^{k}g(x))$$ $$=t^{k}(t^{k})^{k}f(g(x))$$ $$=t^{k(k+1)}f(g(x))$$

So $f(g)$ is homogeneous of degree $k(k+1)$.

Is this correct?

Answer 1

Work from the inside outwards. So $f(g(tx))=f(t^kg(x))=(t^k)^kf(g(x))$.