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 $$y$$.