Reducibility of polynomials after a polynomial substitution

We prove that for any field $K$ and any polynomial $f\in K[x]$ of degree $d$ which is irreducible over $K$, there exists a polynomial $g\in K[x]$ of degree $d-1$ such that the composition polynomial $f(g(x))$ is reducible over $K$. This answers a corresponding question recently raised by Ulas. We also characterize all quartic polynomials $f\in K[x]$, where $K$ is a field of characteristic zero, for which $f(g(x))$ remains irreducible over $K$ under any quadratic substitution $g\in K[x]$. This characterization is given in terms of $K$-rational points on an elliptic curve of genus $1$. As a corollary, we show that the polynomial $g(x)^4+1$ is irreducible over $\mathbb Q$ for any quadratic polynomial $g\in\mathbb Q[x]$.