Algebra of Functions

A function whose co-domain is $R$ (or a subset of $R$) is a real-valued function. When $f,g$ share the same domain $X$, we can combine their output values using ordinary real-number arithmetic, defining new functions on $X$:

The domain need not be a set of numbers at all -- e.g. if $X$ is a class of students and $f,g$ are their marks in two different tests, $f+g$ is exactly their combined (total) marks function.

Properties (mirroring the field properties of $R$ itself): $(f+g)+h=f+(g+h)$; $f+g=g+f$; $0+f=f+0=f$ (zero function as identity); $f+(-f)=(-f)+f=0$; $f(g+h)=fg+fh$ (distributivity); $(c_1+c_2)f=c_1f+c_2f$.

Proof of distributivity (representative proof). To show $f(g+h)=fg+fh$, check both sides agree at every $x$ in the common domain:

$\big(f(g+h)\big)(x)=f(x)\,(g+h)(x)=f(x)\big[g(x)+h(x)\big]=f(x)g(x)+f(x)h(x)=(fg)(x)+(fh)(x)=(fg+fh)(x),$ …