Algebra Lesson: Domain and Range

Domain and Range

When working with functions, we frequently come across two terms: DOMAIN & RANGE. What is a domain? What is a range? Why are they important?

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Definition of Domain

Domain: The set of all possible input values (commonly the "x" variable), which produce a valid output from a particular function. It is the set of all real numbers for which a function is mathematically defined. It is quite common for the domain to be the set of all real numbers since many mathematical functions can accept any input.

Consider a simple linear equation like the graph shown, below drawn from the function \(y=\frac{x}{2}+10\). What values are valid inputs? It's not a trick question -- every real number is a possible input! The function's domain is all real numbers because there is nothing you can put in for x that won't work. That's why the graph extends forever in the x directions (left and right).

Domain

What kind of functions don't have a domain of all real numbers? Well, if the domain is the set of all inputs for which the function is defined, then logically we're looking for an example function which breaks for certain input values. We need a function that, for certain input values, does not produce a valid output, i.e., the function is undefined for that input. Here is an example:

$$ \huge y=\frac{3}{x-1} $$

This function is defined for almost any real x. But, what is the value of y when x=1? Well, it's \(\frac{3}{0}\), which is undefined. Therefore 1 is not in the domain of this function. All other real numbers are valid inputs, so the domain is all real numbers except for x=1.

What other kinds of functions have domains that aren't all real numbers? Certain "inverse" functions, like the inverse trig functions, have limited domains as well. Since the sine function can only have outputs from -1 to 1, its inverse can only accept inputs from -1 to 1. The domain of inverse sine is -1 to 1. However, the most common example of a limited domain is probably the divide by zero issue. When asked to find the domain of a function, start with the easy stuff: first look for any values that cause you to divide by zero. Remember also that we cannot take the square root of a negative number, so keep an eye out for situations where the radicand (the "stuff" inside the square root sign) could result in a negative value. In that case, it would not be a valid input so the domain would not include such values.

Definition of Range

Range: The range is the set of all possible output values (commonly the variable y, or sometimes expressed as f(x)), which result from using a particular function.

The range of a simple linear function is almost always going to be all real numbers. A graph of a line, such as the one shown below on the left, will extend forever in either y direction. There's one notable exception: y=constant (like y=4 or y=19). When you have a function where y equals a constant, your graph is a horizontal line. In that case, the range is just that one value. Otherwise, the range of a linear function is all real numbers.

range

Many other functions have limited ranges. While only a few types have limited domains, you will frequenty see functions with unusual ranges. Here are a few examples:

limited range

As you can see, these two functions have ranges that are limited. No matter what values you enter into a sine function you will never get a result greater than 1 or less than -1. No matter what values you enter into \(y=x^2-2\) you will never get a result less than -2.

Summary: The domain of a function is all the possible input values for which the function is defined, and the range is all possible output values.

domain and range

If you are still confused about domain and range, you might consider posting your question on our message board, or reading another website's lesson on domain and range. Or, you can use the calculator below to determine the domain and range of ANY equation:

Domain and Range Calculator