The position vector is $\vec{r}=(x,y,z)$.
$\vec{A}$ is a vector field, $\vec{A}=(a_1,a_2,a_3)$
and $C$ is a constant.
We have the function
$$f=f(a_1x,a_2x,a_3x,Cx^2).$$
Can one then say, that because axes can be renamed, we can write the arguments $f$ as
$$f=f(a_1y,a_2y,a_3y,Cy^2), or$$ $$f=f(a_1z,a_2z,a_3z,Cz^2)?$$
Should be correct, right? But can we then write $f$ in vector notation
$$f=f(\vec{A}\cdot\vec{r},C\vec{r}^2)?$$
I am unsure if the last line is correct, because it kind of replaces $f:R\rightarrow R$ with $f:R^3\rightarrow R$?
Thus, the question is,
Is this expression correct: $f=f(\vec{A}\cdot\vec{r},C\vec{r}^2)?$