# Discrete & Continuous Domains: Definition & Examples

Instructor: David Karsner

David holds a Master of Arts in Education

All values of x that a function can use is the domain of the function. Domains can be either discrete (a set of values) domain or continuous (over an interval).

## Discrete and Continuous Domains and The Internet

If you have ever filled out an online form, then you have probably encountered both discrete and continuous domains. For example, when ordering something online, you usually have to enter an address. You can pick which state you live in from a drop-down menu. This is an example of a discrete domain. It is discrete because there are only and exactly 50 states to pick from. Meanwhile, if you enter a dollar amount, you are entering a value from a continuous domain. The dollar amount is continuous because any amount from zero to infinity can be entered, with no unacceptable values in between. Let's explore the difference between discrete and continuous domains with examples of both.

## Domain of a Function

A function is a process that takes an input value and maps it to only one output value. The input value is usually represented by x and the output value is represented by y. The domain of the function is a collection of all x-values that can be used. Sometimes you can use all real numbers for the value of x. Other times only a certain interval of numbers can be used; other times only a finite set of x-values can be used. The input values (the x's, the domain) are plugged into a process (the mapping, the function) and the result are the output values (the y's, the range).

#### Examples of Domains

• f(x)=2x+1 : Domain would be all real numbers
• f(x)= ln(x+1) : Domain would be all real numbers greater than -1.
• In a chart of the total rainfall per month of the last year, the domain is the 12 months of the year.

## Continuous Domain

For something to be classified as continuous, it must not contain any gaps over a specific interval.

A continuous domain means that all values of x included in an interval can be used in the function. If the domain of a function was the interval from 1 to 2, that would mean that all values between 1 and 2 (such as 1, 1.232, 1.664324, 3/2, 1.1156, 2, etc.) will work in the function. The values outside of the interval such as -3, 7.1, etc. are not to be included in the domain.

Many functions such as polynomials have a continuous domain that includes all real numbers. The quadratic function f(x)=3x2-2x+3 (also a polynomial) has a continuous domain of all real numbers. Any number whether it be a natural, integer, rational, or irrational can be plugged into this function and map to a y-value.

Some functions have a continuous domain that consist of all real numbers but with a few exceptions. Rational functions typically have a polynomial function in the denominator. You cannot have a zero in the denominator a function. Therefore, any value of x that causes the denominator to equal zero cannot be part of the domain.

#### Example #1

Consider the function y = (2x-5)/(x-3). The (x-3) term in the denominator will equal zero when x=3, since 3-3=0. The domain of this function will thus be all real numbers that don't equal 3.

Some functions have a continuous domain but only on a portion of the x-axis. Take for example the natural log function ln(x). You cannot take the natural log of a negative number. Any x-value that creates a negative for the natural log is not part of the domain. These types of functions create an interval in which the function will work.

#### Example #2

Given the function y = ln(x-5), any value of x of 5 or above will create a positive number or zero inside of the parentheses. Those are the numbers that can be used in the function. The domain of this function would x > or = 5.

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