Set-Theoretic View of Function Composition

Of course we already know of functions $f : A \to B$. Another way of writing this is $$A \stackrel{f}{\to} B.$$ This emphasizes that $f$ is in fact a map taking elements of $A$ to elements of $B$.

Now let's say we have two functions $f : A \to B$ and $g : B \to C$.

Using our notation earlier, we can write this as $$A \stackrel{f}{\to} B \stackrel{g}{\to} C.$$ We can then define the function composition $g \circ f$ by simply taking each element in $A$ and sending it through $f$ and $g$. Notice that $g$ is applied second in this notation.

In other words, $(g \circ f)(x) = g(f(x))$ for every $x \in A$.

Finally, let's try computing a composition ourselves.

That's all there is to it! If you want to get really fancy with a whole chain of compositions, check out commutative diagrams.

You're probably sick of sets of real numbers and functions on real numbers. So if you want, let's take a break from that. (But otherwise, see Evan's alternative explanation.)

Let $Goku\colon \mathbb{R} \to \{\text{English phrases}\}$ be the function defined by $f(x) = ``\text{It's over }\lfloor{x}\rfloor-1\text{.''}$ for real numbers $x$, where $\lfloor{x}\rfloor$ is the floor function.

Let $B$ be the set of boxes (or rectangles) on the Navigator. Let $f\colon B\to \mathbb{R}$ be the constant function defined by $f(b) = 9001$ (or $b\mapsto 9001$ in more suggestive notation) for all boxes $b\in B$.

Let $TP$ be the set of topic pages, like this one or this one. There is a map/function $N.C.\colon B\to TP$ (short for "navigator click") that takes you from a box on the Navigator to the corresponding topic (that's one of the ways you could've gotten to the topic page you're currently on).

Let $E$ be the set of explanations on the whole site (not just the current topic), like this one on dividing complex numbers, or the one you're reading right now. There is a map/function $F.E.\colon TP\to E$ (short for "find explanation") that, given a topic page (like the page you're on right now), shows you a single explanation by default (like the explanation you're reading now).

What does it mean to "compose" two functions? It just means we apply one and then the other (the order is important), at least when it makes sense.

Great! The only two valid compositions using the functions defined above are $Goku\circ f$ (notation for first applying $f$ and then applying $Goku$) and $F.E.\circ N.C.$ (first applying $N.C.$ and then $F.E.$). Actually, valid compositions are functions themselves: $Goku\circ f$ ultimately maps $B$ to the set of English phrases (with the set $\mathbb{R}$ of real numbers as an intermediary), and $F.E.\circ N.C.$ ultimately maps $B$ to the set $E$ of explanations (with the set $TP$ of topic pages as an intermediary).