I'm looking for the name of a function $f(x_1,..,x_n)$ , in which we replace some of the variables $x_i$ by constants.
In such a function, either the constants or the non-constants are often denoted simply as $\cdot$, i.e. $f(x,\cdot)$.
Especially, I'm hoping that there's a definition with which I can find useful theorems that use these functions, e.g.:
Let $f_1(x_1,...,x_n) = f_2(x_1,..,x_n)$ be an equation. Then we can apply any injective function $f$ with one variable on both sides without altering the result.
However, this doesn't cover much: If we e.g. want to add $x_i$ to both sides, then such an $f$ probably won't exist. Instead we use $f'(x,y):= x+ y$, and get the equivalence of the equality
$$
f'(f_1(x_1,...,x_n),x_i) =f'( f_2(x_1,..,x_n),x_i)
$$
, because $f'$ is always bijective if we fix one of the variables.
Its called parametrization. If you have a function $f:\Bbb R^n\rightarrow\Bbb R$ and real numbers $a_1,\ldots,a_m$ with $1\leq m\leq n$, then $f':\Bbb R^{n-m}\rightarrow\Bbb R$ defined by
$f'(x_1,\ldots,x_{n-m}) = f(x_1,\ldots,x_{n-m},a_1,\ldots,a_m)$
is the parametrization of $f$ w.r.t. $a_1,\ldots,a_m$.