Product property for reversing coefficients for polynomials
Hint $\,\ f(P(x))\, =\, x^n P(x^{-1})\ $ is multiplicative, being the product of two multiplicative functions, namely $\,P(x)\mapsto P(x^{-1}),\,$ and $\,P(x)\mapsto x^{\,\deg P}\ $ (an "exponential" of the additive degree map)
Here is an easy way.
Suppose $x \neq 0$, and $P(x) = p_0+...+p_n x^{n_p}$. Then $x^{n_p} P({1 \over x}) = p_0x^{n_p}+p_1 x^{n_p-1} +...+p_{n_p} = f(P)(x)$.
Then $f(P\cdot Q)(x) = x^{n_p+n_q}(P \cdot Q)({1 \over x}) = x^{n_p}(P )({1 \over x}) x^{n_q}(Q)({1 \over x}) = (f(P) \cdot f(Q))(x)$.