Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$.
What do we call $z$ with respect to $y$? Projection, level set, contour, slice, shadow etc?
And also what is $y$ with respect to $z$? A parent, hyper, mother, etc function!
What do we call $g$?
The function given by $z$ is very close to a restriction of the function $f$ (the function given by $y$) to the set where $x_n=g(\ldots)$. $f$ is essentially an extension of this function.
The reason I say "very close" and "eessentially" is that an actual restriction would have to still take $n$ inputs (even if there were only one allowed $x_n$ for each choice of the other inputs). But I would informally call this case a restriction, too - especially if I were imagining things In terms of a fixed $n$-dimensional space (or $n+1$, for the graphs).