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.