# Defining functions and finding functional values

First let's review some definitions.

## Defining a function

A relation is a set of ordered pairs. A function is a relation where each $x$ value is paired with only one $y$ value. (No $x$ value is repeated.) It is sometimes useful to think of the $x$ values as inputs to a function machine and the $y$ values as the output.

### Examples:

$(0,3) \, (5,8) \, (-3,0) \, (8,11) \, (1,4)$ This relation is a function because each value of $x$ is paired with only one $y$ value.
$(3,9) \, (3,-9) \, (0,0) \, (1,-1) \, (1,1)$ This relation is not a function because an $x$ value may be paired with more than one $y$ value, e.g.,when $x$ is $3$ there are two possible $y$ values, $9$ and $-9$.

One way to determine if an equation is a function is to generate some ordered pairs by substituting in various values for $x$ to determine whether each $x$ value is paired with only one $y$ value.

For the equation: $y=2x+5$, let's calculate the $y$ value(s) for $x=3$.

\begin{align} y &= 2x+5 \\ y &= 2(3)+5 \qquad \text{substitute 3 for x} \\ y &= 6 + 5 \\ y &= 11 \\ \end{align}

When $x = 3$, there is only one value for $y$, $y = 11$. This is the point $(3,11)$, with an input value of $3$ and an output value of $11$. For every input value we substitute in for $x$, we get only one output value for $y$.

### More definitions

The special relationship between the input and output values where no input value is repeated--or paired with more than one output value--is called a function. Functions have specific notation. In functional notation, where $y$ is a function of $x$, we replace $y$ with "$f(x)$" which we refer to as "f of x". Each value that we input for $x$ determines one value for $y$.

Written in functional notation, $y = 2x + 5$ becomes $f(x) = 2x + 5$.

A function of $x$ is written as "$f(x)$". This is the output value we get from the input value of $x$.

### Continuing with example $y=2x+5$

Recall that when $x=3$ in the equation $y=2x+5$, $y$ evaluates to 11. That is:

$f(3) = 2(3)+5 = 11$

When we substitute $3$ in for $x$, we get $11$. So, $f(3) = 11$ which means that the function evaluated at $3$ is $11$. This is the point $(3,11)$. (We would say f of 3 is 11.)

But evaluating an equation at only one point is not enough to conclude it's a function. Let's look at a few more points.

When we substitute $-1$ in for $x$, we get $3$.

$f(-1) = 2(-1)+5 = 3$

So, $f(-1) = 3$ which means that the function evaluated at $-1$ is $3$. This is the point $(-1, 3)$. (We would say f of -1 is 3.)

When we substitute $4$ in for $x$, we get $13$.

$f(4) = 2(4)+5 = 13$

So, $f(4) = 13$ which means that the function evaluated at $4$ is $13$. This is the point $(4,13)$. (We would say f of 4 is 13.)

Given the pattern, we conclude that $y=2x+5$ is a function--$y$ is a function of $x$--because no matter what value we substitute in for $x$, we get only one value for $y$.