Function Concept

What Makes a Relation a Function

A function assigns exactly one output to every input in its domain. If $f$ maps a set $A$ to a set $B$ , we write:

f:A\to B

For each $x\in A$ , there must be one value $y\in B$ with $y=f(x)$ . Two different inputs may share an output. What is forbidden is one input receiving two different outputs.

For example, the rule $f(x)=x+2$ assigns one value to every real input:

Input $x$	Output $f(x)$
$3$	$5$
$-1$	$1$
$0$	$2$

The same information can be written as ordered pairs:

\{(3,5),(-1,1),(0,2)\}

Each first coordinate appears with only one second coordinate, so these pairs describe a function on the displayed inputs.

Reading Function Notation

In the equation $f(x)=y$ :

$f$ names the function.
$x$ is an input from the domain.
$f(x)$ is the output produced by applying the rule to $x$ .
$y$ is another name for that output.

The notation $f(3)$ does not mean $f\cdot3$ . It means “evaluate $f$ at the input $3$ .” For $f(x)=x+2$ :

f(3)=3+2=5

Explore One Rule

The interactive function machine uses $f(x)=2x+1$ . Change the input, predict the output, and then compare your prediction with the displayed result.

Function Machine

Enter a number to evaluate the fixed function rule.

f(x) = 2x + 1

The rule always follows the same two operations: multiply the input by $2$ , then add $1$ . For example:

\begin{aligned} f(-4) &=2(-4)+1 \\ &=-8+1 \\ &=-7 \end{aligned}

Function or Not a Function

Compare these two relations:

R=\{(1,4),(2,4),(3,7)\}

Relation $R$ is a function. Inputs $1$ and $2$ share the output $4$ , which is allowed. Relation $S$ is not a function because input $1$ is paired with both $4$ and $6$ .

Check Your Understanding

These questions check one definition at a time, so answer each one by naming the input, the rule, and the single output that the rule produces.

Let $g(x)=3x-2$ . Find $g(0)$ , $g(2)$ , and $g(-3)$ .
Does $T=\{(-2,5),(0,1),(4,5)\}$ define a function on the displayed inputs? Explain.
Does $U=\{(-2,5),(0,1),(-2,7)\}$ define a function on the displayed inputs? Explain.

Worked Solutions

Substitute each input into the same rule:
$g(0)=3(0)-2=-2$
Yes. Every first coordinate in $T$ has exactly one second coordinate. Reusing the output $5$ does not violate the definition.
No. The input $-2$ is paired with two different outputs, $5$ and $7$ .