How to Find the Domain of Piecewise Functions

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  • 0:07 Master of Your Domain
  • 0:37 Piecewise Function
  • 0:47 Find the Domain of a…
  • 3:54 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe
The domain of any function is all the values that x can be for that function. Piecewise functions are special functions that have different parts with unique rules for each part. This lesson will show you how to find the domain for piecewise functions.

Master of Your Domain

Your domain can be defined as your circle of influence. It is the area where you are king, where you call the shots. Different people have different domains, and they can be large or small. It all depends on the person and their circumstances.

Mathematical functions also have domains or areas where they have influence. The domain of a function is the set of input, or x, values for which the function is defined. There are a couple of ways to determine the domain of a function; they will be described later.

Piecewise Functions

A piecewise function is a function that is broken into two or more pieces. Each of these pieces has its own parameters. Here is an example of a piecewise function:


piecewise function


Find the Domain of a Piecewise Function

As mentioned before, there is more than one way to find the domain of a piecewise function. The first way is by looking at the equations that make up the functions. There are three main things that you need to look for.

1. Look at the restrictions of the function. The restrictions are the x < or x > portions of the function.


piecewise function


In this example, the restrictions are x < 5 and x > 5. If there are any places where x is not defined, they are not included in the domain. In this example, x is not ever equal to 5, it is only less than or greater than 5, so the domain of f is all real numbers excluding 5. Written in interval notation, the domain is all real numbers less than or greater than 5.


dom(f)=(-infinity, 5)U(5, infinity)


2. Look at the denominator of any fractions. Since a function is undefined if the denominator of any of the fractions is zero, the domain of that function will exclude any numbers that make the denominator of a fraction zero. Check out this example.


piecewise function


The domain of this function is all real numbers excluding 0 because for the first part of the function, the denominator of the fraction cannot be 0. Since x is the denominator, x cannot equal 0 for this function.


dom(f)=(-infinity, 0)U(0, infinity)


3. Look at any radicals that are present. A function does not work if the number under the square root is negative. So, the domain of the function can not include any numbers that will make the number inside the radical symbol negative. Here is an example:


piecewise function


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