Domain and Range

Learning Objectives

• Define domain and range.
• Identify the domain and range for relations described with words, symbols, tables, sets of ordered pairs, and graphs.

RelationsThe relationship between variables that change together. and functionsA kind of relation in which one variable uniquely determines the value of another variable. describe the interaction between linked variables. These relationships include independent valuesA value or variable that changes or can be manipulated by circumstances. and inputsThe independent variable of a function—input determines output., which are the variables that can be manipulated by circumstances. They also include dependent valuesA value or variable that depends upon the independent value. and outputsThe dependent variable of a function—output is determined by input., which are the variables that are determined by the independent values. There is another pair of components we must consider when talking about relations, called domainThe set of all possible inputs of a function which allow the function to work. and rangeThe set of all possible outputs of a function..

The domain of a function or relation is the set of all possible independent values the relation can take. It is the collection of all possible inputs.

The range of a function or relation is the set of all possible dependent values the relation can produce from the domain values. It is the collection of all possible outputs.

By putting all the inputs and all the outputs into separate groups, domain and range allow us to find and explore patterns in each type of variable.

The domain and range of a function are often limited by the nature of the relationship. For example, consider the function of time and height that occurs when you toss a ball into the air and catch it. Time is the input, height is the output. The domain is every value of time during the throw, and it runs from the instant the ball leaves your hand to the instant it returns. Time before you throw it and after you catch it are irrelevant, since the function only applies for the duration of the toss. Let’s say the ball was in the air for `10` seconds during the toss—in that case, the domain is `0` to `10` seconds. Because time runs continuously during this interval, we can’t write down every possible input, only the starting and stopping values.

The range is every height of the ball during the throw, and it includes all heights between your hand when you let the ball go and the highest point the ball reached before it began to fall back to you. If your hand was `3` feet above the ground when you threw and caught the ball, and the highest it flew was `12` feet off the ground, then the range is `3` to `12` feet. Because height changes continuously during this interval, we can’t write down every possible output, only the starting and stopping values.

Now let’s look at another example of domain and range. Here’s a series of figures each made of groups of squares.

We can make a function out of this series by using the number of the figure as the input, and the number of squares that make up the figure as its output.

An input of `1` has an output of `1`, since figure `1` has just `1` square. An input of `2` has an output of `5`, since figure `2` has `5` squares. An input of `3` has an output of `9`, since figure `3` has `9` squares. This function’s domain is the counting numbers `1`, `2`, `3` that identify which figure is used. The inputs to this function are discrete valuesValues that change in increments (not continuously)., or values that change in increments and not continuously like with the ball-tossing function. There are only the three figures, so the only possible inputs are `1`, `2`, and `3`. Thus the domain of this function is `1`, `2`, `3`. We can enclose the list of values inside curly brackets to indicate that they form a set.

Domain: `{1, 2, 3}`

The range is the number of squares in each figure. The figures have only `1`, `5`, or `9` squares, so that’s the range. There’s no figure that has `2` or `3.5` or any other number of squares. Like the domain, the range is made of a set of discrete values.

Range: `{1, 5, 9}`

We limited the inputs and outputs to `3` each because we were only shown `3` figures. How would the notation for domain and range change if we learned that the pattern goes on indefinitely? Easy! We’d just add a series of ellipses to the end of each set of values, to indicate that the sequence continues, like this:

Domain: `{1, 2, 3,...}`

Range: `{1, 5, 9,...}`

Relations can also be shown as tables or as sets of ordered pairs. Finding the domain and range in these situations is simple, as long as we remember what the terms mean. If a mathematical relationship is given in a table, the independent values, generally listed in the left-hand column, are the domain, and the dependent values, usually found in the right-hand column, make up the range.

The domain can be found by reading down the first column `{-1, 2, 5, 9}`. The range is all the values in the second column `{7, -3, 6, 4}`. 

When it comes to sets of ordered pairs, we simply need to split the pairs apart into `x`-coordinates and `y`-coordinates. Because the `x`-coordinates are the independent values, they make up the domain. The `y`-coordinates are the dependent values, which means that they are the range. Let’s try it.

In the set of ordered pairs `{(-2, 0), (0, 6), (2, 12), (4, 18)}`, the domain is the set of the first number in every pair (those are the `x`-coordinates): `{-2, 0, 2, 4}`. The range is the set of the second number of all the pairs (those are the `y`-coordinates): `{0, 6, 12, 18}`. 

`{(-2, 0), (0, 6), (2, 12), (4, 18)}`

Domain: `{-2, 0, 2, 4}`

Range: `{0, 6, 12, 18}`

We may also encounter functions and relations on graphs. The independent quantity is usually graphed on the horizontal (`x`) axis—that means the `x`-coordinates of the points are the domain. Since the dependent quantity is usually graphed on the vertical (`y`) axis, the `y`-coordinates make up the range. Let’s look at a few graphs to explore how this works.

First, examine this graph of discrete points. The only values that we know to satisfy the function are the marked points. We simply read off the `x`-coordinates, and place them in the set of domain values. Then we read the `y`-coordinates, and put them into the range. For this graph then, the domain is `{-2, 0, 2, 4}`. The range is `{0, 6, 12, 18}`. 

Now let’s look at a different kind of graph, in which the function is a continuous line, which goes on indefinitely in both directions. That means an infinite number of values are part of the function. For this function, there are no restrictions to the domain and range. Any real number can be an input or an output. That means all whole numbers, integers, fractions, and other rational numbers, even irrational numbers, are all part of the domain and part of the range. Since we can’t write down all those possibilities, we simply say that both the domain and the range are all real numbers.

In some situations, only the domain or the range is unrestricted, not both. Consider the graph of the absolute value function, `y = |x|`. The line extends indefinitely in both directions along the `x`-axis, so the domain is all real numbers. However, since the absolute value function transforms any negative value into a positive value, there are no negative numbers in the range. The range is all real numbers greater than or equal to `0`—still way too many to write down.

Functions can be defined using words, symbols, graphs, tables, or sets of ordered pairs, but in each case the parts are the same. The domain is the input, the independent value—it’s what goes into a function. The range is the output, the dependent value—it’s what comes out. Domain and range may be limited to a few discrete values, or they may include all numbers everywhere, to infinity and beyond.