= Pointwise operation
{title2=$h(x)=F(f_1(x),\ldots,f_n(x))$}
A <pointwise operation> on <functions> applies an operation to their values at the same input. If $f,g:X\to Y$ and $\star:Y\times Y\to Y$ is a <binary operation>, the induced operation on the <function space> is defined by
$$
(f\star g)(x)=f(x)\star g(x)\qquad(x\in X).
$$
The common <domain of a function> makes the two inputs available at each $x$; the closure of the operation in the <codomain> makes the resulting value an element of $Y$ again.
For <real-valued functions>, ordinary <addition> and <multiplication> induce <pointwise addition> and <pointwise multiplication>. If $f(x)=x+1$ and $g(x)=x^2$, then $(f+g)(x)=x^2+x+1$ and $(fg)(x)=x^2(x+1)$. <Function composition> instead feeds one output into another rule: $(f\circ g)(x)=x^2+1$ in this example.
Algebraic identities pass to the pointwise construction by evaluation. For instance, if the operation on $Y$ is <associative>, then $((f\star g)\star h)(x)=(f(x)\star g(x))\star h(x)=f(x)\star(g(x)\star h(x))$ for every $x$, proving equality of the two <functions>. This explains how algebraic structure on values induces structure on a <function space>.
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