What is the proper name for compositions like (f∘g)(x)

Question

Addition, subtraction, multiplication, and division of functions, $(f+g)(x)$, $(f-g)(x)$, $(f×g)(x)$, $(f÷g)(x)$, are fairly common. Is there an established name for such operations on functions using any operator to compose $f$ and $g$, similar to how $(f∘g)(x)$ is called function composition?

In other words, how do we call $(f*h)(x)$ where $*$ denotes a binary operation?

The mathematician Kenneth E. Iverson called this a fork (full APL documentation), but this terminology clearly wasn't accepted (unlike other mathematical terms he invented), because fork doesn't find me anything on Wikipedia or MathWorld.

Answer 1

I do not think that it has its own name (at least one that is widely accepted). I would just "multiply" the functions and specify which "multiplication" is being used. That is mostly based on the fact that one often uses the word "multiplication" for the binary operation of a semigroup/monoid/group etc. So I guess that is just the way to go.

Answer 2

Wikipedia calls this action lifting an operation pointwise, thereby forming a pointwise operation.

From the Wikipedia article "Pointwise":

A binary operation $o: Y × Y → Y$ on a set $Y$ can be lifted pointwise to an operation $O: (X→Y) × (X→Y) → (X→Y)$ on the set $X→Y$ of all functions from $X$ to $Y$ as follows: Given two functions $f_1: X → Y$ and $f_2: X → Y$, define the function $O(f_1,f_2): X → Y$ by

$$(O(f_1,f_2))(x) = o(f_1(x),f_2(x)) \text{ for all } x∈X.$$

Commonly, $o$ and $O$ are denoted by the same symbol. A similar definition is used for unary operations $o$, and for operations of other arity.