Table of Contents
Function notation is a compact way to write and talk about functions using symbols. In this chapter we focus on how to read, write, and use this notation, assuming you already understand the basic input–output idea of a function.
The basic symbol: $f(x)$
The most common way to name a function is with a letter, usually $f$, $g$, or $h$.
- $f(x)$ is read as “$f$ of $x$”.
- $x$ is the input (often called the variable).
- $f(x)$ is the output when the input is $x$.
If a function is described by a rule like “take a number and double it, then add $3$,” we can write:
$$
f(x) = 2x + 3
$$
This means: the output is $2x + 3$ when the input is $x$.
Here $f$ is the name of the function, $x$ is the input, and $2x + 3$ is the expression that tells how to compute the output.
Function notation vs. “$y$”
You have probably seen equations written with $y$, such as:
$$
y = 2x + 3
$$
Using function notation, the same relationship can be written:
$$
f(x) = 2x + 3
$$
They play similar roles:
- $y$ and $f(x)$ both represent the output.
- The expression on the right side is the rule.
We can think of $y$ as another name for $f(x)$:
$$
y = f(x)
$$
Function notation has an advantage: you can name different functions easily, such as $f(x)$, $g(x)$, $h(x)$, instead of always using $y$.
Evaluating a function using notation
To evaluate a function means to find the output for a specific input.
If
$$
f(x) = 2x + 3,
$$
then:
- $f(1)$ means: plug in $x = 1$.
$$
f(1) = 2(1) + 3 = 2 + 3 = 5
$$ - $f(4)$ means: plug in $x = 4$.
$$
f(4) = 2(4) + 3 = 8 + 3 = 11
$$
The pattern is:
- Start with the formula for $f(x)$.
- Replace every $x$ with the given input.
- Simplify.
Evaluating with negative numbers and fractions
If
$$
f(x) = 5x - 2,
$$
then:
- $f(-3)$:
$$
f(-3) = 5(-3) - 2 = -15 - 2 = -17
$$ - $f\!\left(\dfrac{1}{2}\right)$:
$$
f\!\left(\dfrac{1}{2}\right) = 5\left(\dfrac{1}{2}\right) - 2
= \dfrac{5}{2} - 2
= \dfrac{5}{2} - \dfrac{4}{2}
= \dfrac{1}{2}
$$
Evaluating with expressions as inputs
The input does not have to be just a number. It can be an expression.
If
$$
f(x) = x^2 + 1,
$$
then:
- $f(3)$:
$$
f(3) = 3^2 + 1 = 9 + 1 = 10
$$ - $f(a)$:
$$
f(a) = a^2 + 1
$$
(Here we are just replacing $x$ with $a$.) - $f(x + 2)$:
$$
f(x + 2) = (x + 2)^2 + 1
$$
If you expand,
$$
(x + 2)^2 = x^2 + 4x + 4,
$$
so
$$
f(x + 2) = x^2 + 4x + 4 + 1 = x^2 + 4x + 5.
$$
The key idea: whatever is inside the parentheses of $f(\,\cdot\,)$ replaces $x$ everywhere in the formula.
Different function names: $f$, $g$, $h$, etc.
You are not limited to $f(x)$. You can have:
- $g(x)$ read “$g$ of $x$”
- $h(x)$ read “$h$ of $x$”
- $p(t)$ read “$p$ of $t$”
- $C(n)$ read “$C$ of $n$”
The letter in parentheses can also change depending on context:
- $s(t)$ might represent distance $s$ as a function of time $t$.
- $C(n)$ might represent cost $C$ as a function of number of items $n$.
Example:
Suppose
$$
g(x) = x^2 - 4x
\quad\text{and}\quad
h(x) = 3x + 1.
$$
Then:
- $g(5) = 5^2 - 4(5) = 25 - 20 = 5$.
- $h(-2) = 3(-2) + 1 = -6 + 1 = -5$.
Having different function names lets us talk about several relationships at the same time.
Inputs and outputs in word problems
Function notation is useful for describing real situations.
Example: A taxi company charges a base fee of \$4 plus \$2 per kilometer.
We can write:
$$
C(d) = 2d + 4
$$
where:
- $d$ is the number of kilometers (input),
- $C(d)$ is the total cost in dollars (output).
Then:
- $C(0)$ is the cost for going $0$ km:
$$
C(0) = 2(0) + 4 = 4
$$ - $C(5)$ is the cost for going $5$ km:
$$
C(5) = 2(5) + 4 = 10 + 4 = 14
$$
Reading $C(5)$ as “$C$ of $5$” helps you remember: plug in $d = 5$ into the formula for $C(d)$.
Changing the input letter
The letter used for the input is not fixed. The function rule stays the same even if we change the variable name.
If
$$
f(x) = 2x + 3,
$$
we could also write the same function as
$$
f(t) = 2t + 3
\quad\text{or}\quad
f(n) = 2n + 3.
$$
These are all the same rule: “double the input and add $3$.” Only the name of the input changes.
What matters is:
- The function name (here $f$).
- How the output is calculated from the input.
Function notation from tables and graphs
From a table
Suppose you have a table:
| $x$ | $f(x)$ |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 10 |
You can read:
- $f(1) = 4$
- $f(2) = 7$
- $f(3) = 10$
The statement $f(2) = 7$ means: when the input is $2$, the output is $7$.
From a graph
On a graph, $x$ is on the horizontal axis, and $f(x)$ is on the vertical axis.
- $f(2)$ is the $y$-value (vertical value) of the point on the graph where $x = 2$.
- If the graph passes through the point $(2, 5)$, then $f(2) = 5$.
Function notation gives a quick way to talk about such points: “The graph goes through $(2, 5)$” is the same as saying “$f(2) = 5$.”
Special outputs in function notation
Certain special values are described neatly with function notation.
- Zero output (root or zero of a function):
Saying $f(a) = 0$ means that when the input is $a$, the output is $. - Same output for two different inputs:
If $f(1) = f(4)$, it means the outputs at $x = 1$ and $x = 4$ are equal.
On the graph, the points $(1, f(1))$ and $(4, f(4))$ are on the same horizontal level.
The point $(a, 0)$ lies on the $x$-axis.
These ideas will be explored more in later chapters, but the notation itself is what lets us express them clearly.
Common mistakes and how to avoid them
- Confusing $f(x)$ with multiplication
$f(x)$ does not mean $f \times x$.
It is the name of the output when the function $f$ is applied to $x$.
- Forgetting to replace every $x$
If $f(x) = 3x^2 - x$, then
$$
f(2) = 3(2)^2 - 2 = 3 \cdot 4 - 2 = 12 - 2 = 10,
$$
not $3(2)^2 - x$ with some $x$ still left in the expression.
- Mixing up the function name and variable
In $g(t) = 4t - 1$:
- $g$ is the function name.
- $t$ is the input.
- $g(t)$ is the output.
If you see $g(3)$, you plug $t = 3$ into the formula for $g(t)$.
- Changing the rule accidentally
If $f(x) = 2x + 5$, then $f(3)$ must use that same rule:
$$
f(3) = 2(3) + 5 = 11.
$$
You cannot change the rule when evaluating.
Summary of key points
- Function notation uses symbols like $f(x)$ to represent the output of a function $f$ for input $x$.
- $f(x)$ and $y$ both represent outputs; $f(x)$ emphasizes the function.
- To evaluate $f(\text{input})$, replace $x$ with the input in the formula and simplify.
- You can use different letters for both the function name ($f, g, h,\dots$) and the input ($x, t, n,\dots$).
- Function notation works with numbers, expressions, tables, and graphs.
- Reading and writing $f(x)$ correctly is a key skill for later algebra and beyond.