Math

Function Notation

6 min readMedium5-question drill

Function notation looks intimidating — $f(x)$, $f(g(x))$, $f^{-1}(x)$ — but it's just a labeling system that lets you talk about functions cleanly. Once you know what each symbol *means*, the questions become procedural.

Where this lesson fits

Before

Linear Equations→Linear Functions→

Function Notation

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→Quadratic Equations →Exponential Functions

A function is a rule that takes an input and gives back exactly one output. We label functions with letters (usually $f$ , $g$ , $h$ ) and write $f(x) =$ [the rule] to define one.

$f(x) = 2x + 3$ means: whatever you plug in for $x$ , multiply by 2 and add 3.

Reading function notation:

$f(5)$ = the output when you plug 5 in for $x$ .
$f(a + 1)$ = the output when you plug $(a+1)$ in for $x$ . Substitute exactly what's inside the parentheses.
$f(g(x))$ = a composition. Compute $g(x)$ first, then plug that into $f$ .

Reading function notation — the cheat sheet

Notation	Means	Example with $f(x) = 2x + 1$
$f(5)$	Plug 5 in for $x$	$f(5) = 2(5) + 1 = 11$
$f(a)$	Plug $a$ in for $x$	$f(a) = 2a + 1$
$f(x + 2)$	Plug $(x+2)$ in for $x$	$f(x+2) = 2(x+2) + 1 = 2x + 5$
$f(g(x))$	Composition: do $g$ first, then $f$	If $g(x) = x^2$: $f(x^2) = 2x^2 + 1$
$f^{-1}(y)$	Inverse: input $y$, output the $x$ that gave it	$y = 2x+1$ → $x = (y-1)/2$, so $f^{-1}(y) = (y-1)/2$

Plugging in is the key skill. When you see $f(x+2)$ , replace EVERY $x$ in the function's rule with $(x+2)$ — including the parentheses. If $f(x) = x^2 + 3x$ , then $f(x+2) = (x+2)^2 + 3(x+2)$

Composition: $f(g(x))$ . Read it inside-out. If $f(x) = x + 1$ and $g(x) = 2x$ :

Compute $g(x) = 2x$ first.
Plug that into $f$ : $f(2x) = 2x + 1$ .
So $f(g(x)) = 2x + 1$ .

Note: $f(g(x))$ and $g(f(x))$ are usually different! Order matters.

How to compute $f(g(x))$ at a specific value

Have you computed $g$(input) yet?

Yes ↓

Now plug that result into $f$ — done?

Yes ↓

→ Result is $f(g(\text{input}))$ — composition complete

No ↓

→ Plug the $g$-result into $f$'s rule. That's the final answer.

No ↓

→ Start with $g$(input). Inside-out. Then feed that to $f$.

Reading from a graph or table:

"Find $f(3)$ from this graph" → look at $x = 3$ , read off the $y$ -value.
"Find $x$ when $f(x) = 5$ " → look at $y = 5$ , read off the $x$ -value (could be more than one).

Inverse functions $f^{-1}(x)$ : the function that undoes $f$ . If $f(2) = 7$ , then $f^{-1}(7) = 2$ . To find $f^{-1}$ algebraically: write $y = f(x)$ , swap $x$ and $y$ , then solve for $y$ .

Quick check

Quick check. Function notation = substitute. Wrap whatever's inside the parentheses with its own parentheses, then evaluate carefully.

If f(x) = 4x + 2 and g(x) = 4x + 4, what is f(g(4))?

Worked examples

Example 1

If $f(x) = x^2 - 2x$ , what is $f(a + 3)$ ?

Example 2

If $f(x) = 3x - 2$ and $g(x) = x^2$ , what is $f(g(2))$ ?

Common pitfalls

Forgetting parentheses when substituting

$f(x+2)$ with $f(x) = x^2$ is $(x+2)^2$ , NOT $x^2 + 2$ or $x + 2^2$ . Always wrap the substitution in parentheses, expand carefully, then simplify.

Confusing $f(x) \cdot g(x)$ with $f(g(x))$

$f(x) \cdot g(x)$ is multiplication. $f(g(x))$ is composition (plug $g$ 's output into $f$ ). They're completely different operations.

Reading composition outside-in

$f(g(x))$ is read inside-out: $g(x)$ first, then $f$ . Don't compute $f(x)$ and then plug into $g$ — that's $g(f(x))$ , which usually gives a different answer.

Confusing $f(2)$ with $f \cdot 2$

$f(2)$ means evaluate $f$ at input 2 — substitute. $f \cdot 2$ means multiply $f$ by 2 — different. The parentheses indicate function evaluation, not multiplication.

Key takeaways

$f(x)$ is a function — a rule mapping each input to one output. The letter ( $f$ , $g$ , $h$ ) is just the function's label.
Substitution: replace every $x$ in the rule with whatever's inside the parentheses, using parentheses to keep order of operations.
Composition $f(g(x))$ is inside-out: $g$ first, then $f$ . Order matters.
From a graph: $f(a)$ is the $y$ -value at $x = a$ . To find $x$ when $f(x) = b$ , look at $y = b$ and read off the $x$ .
Inverse $f^{-1}$ undoes $f$ : if $f(a) = b$ , then $f^{-1}(b) = a$ .

Watch & learn

Curated Khan Academy walkthroughs on Function Notation. They're complementary to this lesson — watch one if a written explanation isn't clicking, or after to reinforce.

Try it yourself

5 practice questions on Function Notation, drawn from the question bank. The tutor is one click away if you get stuck.

Where this lesson fits

Before

Linear Equations→Linear Functions→

Function Notation

You are here

→Quadratic Equations →Exponential Functions