Is there a technical name for $f$ such that $(g \circ f)(a_1, a_2,\cdots)$ yields $g(a_1, a_2, \cdots)$? Is there a technical name for a function $f$ such that $(g \circ f)(a_1, a_2,\cdots) \rightarrow g(a_1, a_2, \cdots)$? That is, is there a name for a function $f$ such that the result of composing $g$ with $f$ is $g$ invoked with $f$'s arguments?
EDIT: Accidentally reversed the order of operations and put $f \circ g$ when I meant $g \circ f$ (i.e. $g(f(\cdots)$).
 A: In general for a binary operation $\circ$...
If $f \circ g=g$, then $f$ is a left identity for $g$ and if $g \circ f = g$, then $f$ is a right identity for $g$.
If $\circ$ is composition and $f$ is a right identity for all functions $g$, then $f$ must be the identity function: $f(x_1,\dots,x_n)=(x_1,\dots,x_n)$. The same is true if we change "right" to "left".
In fact, if $g$ has a left inverse (which is true iff it is one-to-one), then $\mathrm{id}=g^{-1} \circ g = g^{-1}\circ g\circ f= \mathrm{id}\circ f=f$.
If $g$ is not one-to-one, there are potentially many functions (other than the identity) such that $g\circ f=g$.
When "left" is replaced by "right", "one-to-one" should be replaced by "onto".
A: To add to Bill's answer: in certain contexts we would say that "$f$ is a symmetry of the function $g$" or that "$g$ is invariant under $f$". 
For example, if $g(x) = x^2$ and $f(x) = -x$, we say that $g$ is symmetric under reflection and $f$ is the reflection operation. 
Another example, if $g(x) = \sin x$ and $f(x) = x+2\pi$, we say that $g$ is periodic with period $2\pi$, or that $g$ is invariant under translation by $2\pi$, or that $g$ is invariant under $f$. 
