**How do we find Domains & Ranges**

Let's take a look at an equation that has two unknown variables.

Suppose that we have a cost function of the form

In order to graph or plot this function, we have to give X a fixed
value and substitute X into the equation to find out what the value of Y
would be.

Suppose that we let X be 0, then

Similarly, if we let X be 1, then

If we let X be -3, -2, -1, 0, 1, 2, 3 respectively and subtitute each
value of X into the equation, then the set of ordered pairs would be:

If we use the set of ordered pairs above to plot the graph, the
graph would be:

What is the **domain** and **range** associated with the
given function?

The **domain** for the above function is **the set of all
real numbers**. If we examine the set of ordered pairs, it will include
{..., -3, -2, -1, 0, 1, 2, 3, ...}.

The **range** for the above function is **the set of all
real numbers greater than or equal to -3**. If we examine the set
of ordered pairs, it will include {..., 6, 1, -2, -3, -2, 1, 6, ...}.

**A few simple facts that you should know**

The **domain** of a function is associated with the
**X-axis** of the graph and the **range** of a function
is associated with the **Y-axis** of the graph?
For certain functions, the **domain and range** can be found by simply
looking at the graph. In the above example, the domain is the set of all
real numbers because the head of the arrow indicate that we can continue
in that direction to encompass the whole X-axis. And the range is
the set of all real numbers greater than or equal to -3 because lowest
point of the curve on the Y-axis is -3.