Consider a function $f : \mathcal X \times \mathcal Y \mapsto \mathbb R$. I want to define $g_x(y) = f(x,y) : \mathcal Y \mapsto \mathbb R$. I want to say that
$g_x$ is a ___ of function $f$.
What is the appropriate word for _____
I've seen your $g_x$ called the $x$-section of $f$. E.g. Folland's Real Analysis, section 2.5.
Edit: Another notation that's often useful is to write $f(x, \cdot)$ instead of $g_x$.
Edit: Ah, I was assuming you were varying $x$. If $x$ is fixed, you might want to call $g_x$ a restriction of $f$.