For $n\ge 1$, let $g(x_1,x_2,\ldots,x_n)$ be an irreducible <i>homogeneous</i> polynomial in $n$ variables over a field $k$ and $f(x)$ an irreducible polynomial of $k[x]$. Is $f(g(x_1,x_2,\ldots,x_n))$ necessarily irreducible?

For instance this holds when $n=1$ (since then $g(x_1)=\lambda x_1$), or when $f$ has degree 1 (by a simple argument).