Domain and Range - Definition & Finding Graphically

Domain, Range, and Interval Notation

Generally speaking, the domain of a function is all possible inputs, and the range of a function is all possible outputs. Consider the following two sets: $$\begin{align}\text{Inputs}&=(1, 2, 3, 4, 5)\\ \text{Outputs}&=(6, 7, 8, 9, 10) \end{align}$$ The domain would be the integers $1$ through $5$. The range would be the integers $6$ through $10$.

We can find the domain and range of a function by looking at its graph. In a regular graph, the domain is all of the possible $x$ values, while the range is all of the possible $y$ values. Let's start by looking at the graph of $y=x$.

This graph expands infinitely in both directions and on both axes. This means that both the domain and range include all real numbers. All linear functions that don't specify a stopping point will have a domain and range of all real numbers.

Let's look at a function that has a more defined range, $y=x^2$.

The domain of this graph is still all numbers as any $x$ value can be put into the function, but there is a minimum $y$ value of $y=0$. This means that the range is $0 \le y < \infty$.

Some functions have restrictions on both the domain and range, such as $y=\sqrt{x}$. This is the square root graph or a square root function.

This graph has no $x$ values less than $0$, as well as no $y$ values less than $0$. Let's determine the domain. This means that both the domain and range include all numbers from $0$ to $\infty$. Our domain is a set of numbers greater than and equal to zero.

We're going to try a practice problem to see how we're doing.

The domain of the graph is all real numbers as any $x$ value can be put into our function. The range of the graph is $-1 \le y \le 1$ because all possible outputs will fall in between those two numbers.

Interval Notation

All of our domain and range intervals can be written with interval notation. When using interval notation it is important to remember that parentheses $($ $)$ denote exclusive intervals, or intervals that don't include the outermost numbers, and brackets $[$ $]$ denote intervals that are inclusive and do include the outer numbers. Here is an overview of each of the different intervals. $$\begin{align}[a,b]&:\text{Both outer bounds are inclusive}\\ [a,b)&:\text{The lower bound is inclusive and the upper bound is exclusive}\\ (a,b]&:\text{The lower bound is exclusive and the upper bound is inclusive}\\ (a,b)&:\text{Both bounds are exclusive} \end{align}$$

We can write our domain and range from the examples above in interval notation. $$\begin{align}\text{Inputs}&=(1, 2, 3, 4, 5)\\ \text{Outputs}&=(6, 7, 8, 9, 10) \end{align}$$ The domain is $[1,5]$ and the range is $[6,10]$

For the function $y=x$, our domain and range were both all real numbers. This can be written as $(\infty,\infty)$. Any time one of your bounds is $\infty$, the interval is exclusive as there is no real end to the interval.

Try this one for yourself:

The correct interval notation for both the domain and range is $[0,\infty]$.

Wrap Up

• All functions have a domain and range
• Domain is the entire set of inputs, on a graph this is normally all $x$ values
• Range is the entire set of outputs, on a graph this is normally all $y$ values
• Different types of functions have restrictions on their domain and range which can be seen through their graphs
• Interval notation is used to quickly show the domain and range of a function

(Video) Find the Domain and Range From a Graph

by Mathbyfives

In this video, Mathbyfives provides examples on how to find domain and range from a graph.

Summary

Some important definitions before we begin:

• A relation is any set of ordered pairs.

• The set of all first components of the ordered pairs is called the domain.

• The set of all second components is called the range.

The domain is the set of all $x$ values, so we want to understand the range of all $x$ values that are being represented by a graph. The range is the set of all $y$ values that are represented by a function on a graph.

Example 1

Let's look at the domain for this first example. The domain is the set of $x$ values that the function is represented by. We can see the open circle at $x=3$, and we see a closed circle at $x=4$. So, we can determine that the domain of this graph is $(-3,4]$. Now, let's look at the range. The open circle is at $y=-2$, and the closed circle is at $y=1$, so we can write the range as $(-2,1]$.

Example 2

We can see that there are arrows on both sides of the function, so the graph goes on to infinity. Therefore, the domain is $(-\infty, \infty)$ and the range is also $(-\infty, \infty)$.

Finding Domain and Range Graphically

We can find the domain and range algebraically, but we can also find the domain and range graphically.

The domain is the set of all $x$ values a function can have. We can visually determine the domain by looking at the expanse of $x$ values the function is describing. Take your pencil, and start at one end of the function, that is one end of your domain. Trace the function, taking stock of the values not represented by the function's graph. When you hit the other end of the function, that is the end of your domain.

The range is the set of all $y$ values a function can have. You can use the same technique as you would use for domain, except this time look at the $y$ values.

Example

Let's look at the following graph of the function $y=x^2-4$.

Let's first look at the domain. We can see that the graph seems to go on to infinity on either end of the $y$-axis. If we take our pencil and trace the graph, it will encompass all real $x$ values. Therefore, the domain can be written in interval notation, like so: $(-\infty, \infty)$.

Now, let's look at the range. The function seems to go on to infinity for positive $y$ values, but the same cannot be said for the negative $y$ values. We can see that the lowest $y$ value is at $y=-4$. Therefore, the range can be written in interval notation, like so: $[-4, \infty)$

Looking Further

In the next lessons, we will continue to learn about function operations. We will start with how to add and subtract and multiply and divide functions. We will take that info and learn about composite functions.

So far, we have looked at functions that have fairly standard domains and rangers. When a function has different formulas on different parts of its domain, it is called a piecewise function. We will commonly see these when dealing with step functions.

Another point to note: A function is onto when its range matches its entire codomain.