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 [math]x[/math] value is paired with only one [math]y[/math] value. (No [math]x[/math] value is repeated.) It is sometimes useful to think of the [math]x[/math] values as inputs to a function machine and the [math]y[/math] values as the output.

Examples:

[math](0,3) \, (5,8) \, (-3,0) \, (8,11) \, (1,4)[/math] This relation is a function because each value of [math]x[/math] is paired with only one [math]y[/math] value.

[math](3,9) \, (3,-9) \, (0,0) \, (1,-1) \, (1,1)[/math] This relation is not a function because an [math]x[/math] value may be paired with more than one [math]y[/math] value, e.g.,when [math]x[/math] is [math]3[/math] there are two possible [math]y[/math] values, [math]9[/math] and [math]-9[/math].

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

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

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

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

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 [math]y[/math] is a function of [math]x[/math], we replace [math]y[/math] with "[math]f(x)[/math]" which we refer to as "f of x". Each value that we input for [math]x[/math] determines one value for [math]y[/math].

Written in functional notation, [math]y = 2x + 5[/math] becomes [math]f(x) = 2x + 5[/math].

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

Continuing with example [math]y=2x+5[/math]

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

[math]f(3) = 2(3)+5 = 11[/math]

When we substitute [math]3[/math] in for [math]x[/math], we get [math]11[/math]. So, [math]f(3) = 11[/math] which means that the function evaluated at [math]3[/math] is [math]11[/math]. This is the point [math](3,11)[/math]. (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 [math]-1[/math] in for [math]x[/math], we get [math]3[/math].

[math]f(-1) = 2(-1)+5 = 3[/math]

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

When we substitute [math]4[/math] in for [math]x[/math], we get [math]13[/math].

[math]f(4) = 2(4)+5 = 13[/math]

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

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