Let $f$ and $g$ be a pair of functions mapping reals to reals. It is common to use the point-free notation $f\circ g$ to describe the function $h$ defined by $h(x)$ = $f(g(x))$. By "point-free" I mean the notation $f\circ{g}$ does not refer explictly to any function arguments $x$.

Now consider $f:\mathbb{R}^2\to\mathbb{R}$ with $g:\mathbb{R}\to\mathbb{R}$

Is there a commonly used "point-free" notation for the composite function $h$ defined by $h(x,y) = f(x,g(y))$? That is, could we write $h$ in a generic way without reference to $x$ and $y$?

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    $\begingroup$ You could write $h:=f\circ({\rm id},g)$. $\endgroup$ – Christian Blatter Jun 22 '18 at 15:59
  • $\begingroup$ Sometimes $($id$,g)$ is written as id$\times g$. $\endgroup$ – Michael Burr Jun 22 '18 at 16:07

Write $(\pi_1,g \circ \pi_2)$ for the function: $$ (\pi_1,g \circ \pi_2): \mathbb{R}^2 \to \mathbb{R}^2: (x,y) \mapsto (\pi_1(x,y),g(\pi_2(x,y))) = (x,g(y)). $$ Then we have $h = f \circ (\pi_1,g \circ \pi_2)$. Usually with this kind of compositions, the projection functions $\pi_1, \pi_2$ can be useful.


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