Foundations — Composition of functions — f(g(x)), g(f(x))

Before you can compose functions, you must be fluent in the little pieces the parent note throws around: arrows, letters like $f$, the mysterious circle $\circ$, "domain", "range", nested parentheses. We build them one at a time, and each one earns its place before it is used again.

1. The function machine: $f(x)$

Read $f(x)$ out loud as "f of x". It does not mean "$f$ times $x$". The brackets are not multiplication — they are a feeding slot: whatever sits inside them is the thing being dropped into the machine.

Figure s01 — a rounded box labelled $f$ (the machine) with a magenta arrow feeding the input $x=3$ into its slot on the left, and an orange arrow carrying the output $f(3)=10$ out on the right; the rule "square it, add 1" is printed inside the box.

• The name $f$ (or $g$, or $h$) labels which machine we are using.
• The input is whatever is inside the brackets.
• The output is what comes out — the value $f(x)$ stands for.

2. The arrow: $f:A\to B$

The parent writes $f:A\to B$ and, using a second machine named $g$ (Section 1 — any letter names a machine), $g:B\to C$. Let us earn every piece.

The $\to$ arrow is chosen because a function has a direction: things flow in one way. You feed $A$, you never feed $B$. This one-way arrow is exactly why order will matter later.

Figure s02 — three white circles labelled $A$, $B$, $C$ left to right; a violet arrow marked $f$ runs from $A$ to $B$, an orange arrow marked $g$ runs from $B$ to $C$, showing the machines lined up head-to-tail so that $B$ is both the out-pile of $f$ and the in-pile of $g$.

Why does the topic need this? Because composition demands that the out-pile of the first machine fits into the in-slot of the second. The parent's rule "the outputs of $f$ must be a subset of the inputs $g$ accepts" is just: the arrows must line up head-to-tail, like $A\to B\to C$.

3. Domain and range: the in-pile and the out-pile

So we now have three named piles: the domain ($A$, inputs), the codomain ($B$, declared outputs), and the range (the part of $B$ genuinely hit). See Domain and range of function for the full story. Here is the picture that matters for composition:

• $f(x)=\sqrt{x}$ refuses negative inputs — you cannot square-root $-4$ and stay in real numbers. Its domain is $x\ge 0$.
• $g(x)=x-4$ accepts anything — its domain is all real numbers.

4. Nested parentheses: innermost first

The parent constantly writes things like $g(f(x))$. This is a nesting, and there is one iron rule.

This is the same rule you already use in arithmetic: in $g(f(5))$ you compute $f(5)$ before $g$, exactly as in $2\times (3+4)$ you compute $3+4$ before multiplying. Innermost bracket wins.

Figure s03 — the expression $g(f(x))$ printed large; a magenta box rings the inner $f(x)$ labelled "compute this inner value first", and a violet downward arrow leads to the caption "drop that value into g's slot", with the arithmetic analogy $2\times (3+4)$ underneath.

Why the topic needs this: the entire meaning of $g(f(x))$ depends on reading it inside-out. Read it outside-in and you would apply $g$ first — which is the wrong machine and the wrong answer.

5. The circle $\circ$: the composition symbol

Now we can meet the star of the topic.

The circle is just a compact label for "the assembly-line machine". Everything the parent proves — non-commutativity, associativity, the identity — is a fact about this combined machine.

6. The identity machine $I(x)=x$

Why does the topic need a do-nothing machine? Because composition needs a neutral element — something that, when bolted on, changes nothing. It plays the role that $0$ plays for adding and $1$ plays for multiplying: $f\circ I=f$ and $I\circ f=f$ (where $I$ is taken on whichever set makes the arrows line up). It also secretly powers Inverse functions, where a function and its inverse compose back to $I$.

How the foundations feed the topic

The map below reads top-to-bottom (graph TD = "graph, top-down"): an arrow $X\to Y$ means "you need $X$ before $Y$ makes sense." Trace it: the function machine (top) is the seed of everything. It grows two shoots — the arrow notation (which sets $A$ and $B$) and the rule about nested brackets. The arrow notation feeds domain and range; nested brackets and domain/range together feed the circle symbol. The machine also spawns the identity. Finally, the circle, the identity, and domain/range all pour into full composition (the box near the bottom), which then flows onward to inverses and the chain rule.

Each box is a symbol or idea you now own. The bottom box — actual composition — is what the parent note runs with. From there the path continues to Inverse functions, Bijective functions, and Derivatives - chain rule, all of which lean on this same machinery.

Equipment checklist

Cover the right side and test yourself — if you can answer all of these, you are ready for the parent note.