*Contributor: Ashley Nail. Lesson ID: 13666*

Do you know how to find the domain and range of functions? Look at the graph or table, and you will quickly see the relationship between sets of numbers. Find out how!

categories

subject

Math

learning style

Kinesthetic, Visual

personality style

Lion, Otter, Beaver, Golden Retriever

Grade Level

High School (9-12)

Lesson Type

Quick Query

Jenna is a mechanic, and she needs to order tires for her shop. She wants to make a set of numbers that she can look at any time to determine how many tires to order each week.

- How can we use our knowledge of functions, domains, and ranges to help Jenna?

A function is the relationship between the set of numbers in a domain to the set of numbers in a range.

We also know that each member of the domain relates to exactly one member of the range. In other words, a member of the domain cannot be connected to two values in the range.

This is why the vertical line test proves functions.

If you were to input values into a function and gather a set of output values, you would have a domain and range. You could put those values in a table, then make ordered pairs, and lastly graph the sets of numbers.

As long as you could run a vertical line left to right across the graph and as long as that line never hits more than one point at a time, you have a function!

If you need more review, visit our lesson found under the **Additional Resources** in the right-hand sidebar.

Now, it is time to learn how to name domain and range sets by looking at these tables, graphs, and equations.

Let's take the example from above:

After inputting values to the function, we have a set of output numbers. These two sets make up the domain and range of this function.

Each member of the domain relates to exactly one member of the range. Every time you input 4, you will always get 9 and no other value.

Now, we can organize this data in a table and then make ordered pairs:

If the ordered pairs and the table were the only data we had, we could name the domain and range.

The domain is the set of inputs. We notate these sets using brackets { } to contain the numbers:

Domain: {1, 4, 6, 7, 10}

The range is the set of outputs. We notate those the same way:

Range: {3, 9, 13, 15, 21}

Notice that the domain and range sets are always listed from least to greatest.

However, with this function, we have more information. We have the equation f(x) = 2x +1.

That means these are not complete domain and range sets. Let's look at the graph.

If we graph the ordered pairs, our coordinate plane looks like this:

But we know we can input even more values into this function.

For example, we could input -2 and get the output value of -3. This means -2 is also a member of the domain, and -3 is a member of the range.

Go ahead and input more values into this function.

- What do you quickly notice?

We could input values all day long. We could input positive and negative integers. We could input fractions and decimals and continue to get output values.

This function makes a linear graph:

Now, we need to update our domain and range.

Looking at this graph, we notice the x-value, or input, can be all real numbers. Therefore, the domain is "x such that x is a member of all real numbers."

Looking at the graph, we also notice the y-value, or output, can be all real numbers:

Let's look at another graph:

If you look at the x-values, the graph starts at 1 and continues on forever, including all numbers greater than or equal to 1.

This means the domain for this graph is {x | x ≥ 1}

If you look at the y-values, the graph starts at -4 and continues on forever, including all numbers greater than or equal to -4.

This means the range for this graph is {y | y ≥ -4}.

- Ready to practice naming the domain and range sets for functions?

Click NEXT to visit the *Got It?* section to practice naming functions using tables and graphs.