Intuition The one core ideaA function is a machine with a name : you drop a number in one end, it follows a fixed rule, and exactly one number comes out the other end. Everything about function notation — f x f ( x ) g ( f ( x )) which machine , what went in , and what came out .
Before you can trust the parent note on Function notation — f(x), g(x) , you need every piece of its alphabet built from nothing. This page collects each symbol and idea it uses or silently assumes , and defines them one at a time, each resting on the one before.
Definition What a number "is" here (the picture)A number is a position on a straight line. The line stretches left (negative) and right (positive), with 0
This is the very first picture we need, because a function machine eats a number (a position on the line ) and produces another number (another position ).
Why the topic needs it: when we later write f ( 2 ) f ( − 1 ) 2 − 1
A variable is a named empty box that will hold a number later. We usually call it x x
Intuition Why a letter instead of a number?If I want to describe a rule that works for every input, I can't write it out for 1 2 3 x
Why the topic needs it: function notation separates the rule from the specific number . The variable x
An expression is a recipe built from numbers, the box x + − × 2 x + 1 2 1
Why the topic needs it: the parent note's rules like f ( x ) = 2 x + 1 are expressions. The expression is the "recipe card" hanging inside the machine.
This is the single most important action in the whole topic, so it gets its own picture.
Substitution means: erase the box x , then compute.
Intuition Why "everywhere"?If the recipe mentions the box three times, all three copies are the same box, so they all get the same number. Miss one, and you've mixed two different inputs — that's a broken machine.
Common mistake The classic slipWith f ( x ) = 3 x 2 − 2 x + 5 f ( − 1 ) both x 3 ( − 1 ) 2 − 2 ( − 1 ) + 5 x whole value, brackets and all.
Why the topic needs it: every evaluation (f ( 2 ) g ( a + 1 ) f ( g ( 2 )) is a substitution. Master this and the rest is bookkeeping.
Definition Two meanings of=
Defining: f ( x ) = 2 x + 1 sets up the machine — it announces the rule. Read it "f 2 x + 1 Computing: f ( 2 ) = 5 reports a result — it states what came out.
Same symbol, two jobs: making the machine vs. reading its output.
Why the topic needs it: confusing these two is the root of the parent's "Common Mistake 2" (writing f = x 2 + 1 rule and number separate.
Now we can assemble the star of the show.
Definition Function (as a machine)A function is a machine that takes one input number and returns exactly one output number, always following the same rule.
Name — a letter like f g h which machine.Input — the number you feed in (lives in the box x Rule — the expression inside (e.g. 2 x + 1 Output — the number produced, written f ( x )
Intuition "Exactly one output"One input, one output — no menus, no maybes. Feed in 3 always get the same answer back. This is what makes it a function and not just a vague "some result."
Why the topic needs it: this machine picture is the mental model behind literally every callout in the parent note.
f ( x ) f ( x ) f x the output when machine f x ".
f ( ) x
So f ( 2 ) f 2
f ( x ) not f x not mean multiply. f ( 3 ) f 3 f × 3
Why the topic needs it: this is the topic. Everything else is what you can do once you can read f ( x )
Definition Why more than one letter?We use different names (f g h different machines . Then f ( 5 ) g ( 5 ) rules differ.
Worked example Same input, different machinesWith f ( x ) = 2 x − 3 g ( x ) = x 2 + 1 f ( 5 ) = 2 ( 5 ) − 3 = 7 , g ( 5 ) = 5 2 + 1 = 26. 5 f g
Why the topic needs it: comparing and combining functions (the parent's whole "power of notation" section) is impossible without distinct names.
Definition Composition (inside-out)Writing f ( g ( x )) run g f You always work from the innermost bracket outward.
Intuition Why inside-out?f has an input. That input is g ( x ) g g
Worked example Order mattersWith f ( x ) = 2 x g ( x ) = x + 3 f ( g ( 5 )) = f ( 8 ) = 16 , g ( f ( 5 )) = g ( 10 ) = 13. 16 = 13 Composite functions .
Why the topic needs it: composition is the reason notation exists — it lets us stack machines unambiguously.
Substitution put a value in
Function machine name rule output
Read it downward: each idea only makes sense once the ones feeding into it are solid.
Functions: Domain and Range — which inputs the box is allowed to hold.Composite functions — the full machinery of f ( g ( x )) Inverse functions — the machine that undoes f Graphing functions — drawing f ( x ) Piecewise functions — machines with more than one rule.
Give an out-loud answer, then reveal.
Where does a number "live" in our picture? As a marked position on a straight number line.
What is the variable x A named empty box that will hold any number you choose later.
What is an expression like 2 x + 1 A recipe of operations on the box — instructions, not yet a number.
What does "substitute" mean? Erase x
The two jobs of the = Defining the machine (f ( x ) = 2 x + 1 f ( 2 ) = 5
What are the four parts of a function machine? Name, input, rule, output.
How do you read f ( 3 ) "f 3 f 3
Does f ( 3 ) f × 3 No — the bracket means "feed in," not multiply.
Why use different letters f g They name different machines, so the same input can give different outputs.
In f ( g ( x )) g f
Is f ( g ( x )) = g ( f ( x )) No — order matters; composition is generally not commutative.
