Functions

Definitions

A function is like a machine that follows a specific rule. For every number you put in, you get exactly one number out.
Let's imagine a machine whose rule is "multiply by 2".

If we put in $3$, we get out $6$. If we put in $5$, we get out $10$. A table of values helps us organize these input–output pairs:

Input	3	5	8	10
Output	6	10	16	20

To work with these rules more efficiently, mathematicians use a special notation.
To represent this machine, we write $\textcolor{olive}{f}(\textcolor{colordef}{\text{input}}) = \textcolor{colorprop}{\text{output}}$. The parentheses $($ $)$ indicate that the function $\textcolor{olive}{f}$ is applied to the input.
We use function notation to name functions and their variables, replacing “$\textcolor{colordef}{\text{input}}$” by “$\textcolor{colordef}{x}$” and “$\textcolor{colorprop}{\text{output}}$” by “$\textcolor{colorprop}{f(x)}$”.
For example, if the rule is “twice the input”:

we have $\textcolor{olive}{f}(\textcolor{colordef}{x}) = 2x$.
When the input is $\textcolor{colordef}{x} = \textcolor{colordef}{1}$, we get:$$\begin{aligned}\textcolor{olive}{f}(\textcolor{colordef}{1}) &= 2 \times \textcolor{colordef}{1}\\ &= \textcolor{colorprop}{2}\end{aligned}$$

Definition Function

A function is a rule that assigns to each input value from a set called the domain exactly one output value in a set called the range.
We use the notation $f(x)$ to represent the output of a function $f$ when the input is $x$.

$f$ is the name of the function (the rule).
$x$ is the input variable.
$f(x)$ is the output value when the input is $x$. It is read as “$f$ of $x$”.

Example

The function $f$ is defined by the rule $f(x)=2x-1$. Find the value of $f(5)$.

Answer

To find $f(5)$, we substitute the input value $x=5$ into the function's rule:$$\begin{aligned}[t]f(x) &= 2x - 1 \\ f(5) &= 2(5) - 1 \\ &= 10 - 1 \\ &= \boldsymbol{9}\end{aligned}$$

Tables of Values

Definition Table of Values

A table of values is a table that organizes the relationship between the input values ($x$) and their corresponding output values ($f(x)$) for a function.

Example

Complete the table of values for the function $f(x)=x^2$.

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$

Answer

We substitute each value of $x$ into the function $f(x)=x^2$:

$\begin{aligned}[t] f(-2) &= (-2)^2 \\ &= 4 \end{aligned}$
$\begin{aligned}[t] f(-1) &= (-1)^2\\ &= 1 \end{aligned}$
$\begin{aligned}[t] f(0) &= (0)^2 \\ &= 0 \end{aligned}$
$\begin{aligned}[t] f(1) &= (1)^2 \\ &= 1 \end{aligned}$
$\begin{aligned}[t] f(2) &= (2)^2 \\ &= 4 \end{aligned}$

The completed table is:

$x$	$-2$	$-1$	$0$	$1$	$2$
$f(x)$	$4$	$1$	$0$	$1$	$4$

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)	\(4\)	\(1\)	\(0\)	\(1\)	\(4\)