Function  Essentials What is a Function? Identify the type of function that you are dealing with. The strategies for finding the domain of a function depend on the characteristics of the function. When you first begin learning about functions, you may be given example problems that simply list a set of points (that is, x-values and their corresponding y-values). For these types of functions, the domain is just the list of x-values for each of the points.

How To Find the Domain of a Function Learn the definition of the domain. Before you can begin finding the domains of specific functions, you must first have a strong understanding of what exactly the domain is. The domain is defined as the set of input values for which the function produces an output value. In other words, the domain is the full set of x-values that can be plugged into a function to produce a y-value.

 

Set  Notation    

Set-­‐Builder  Notation  

  Here is a simple example of set-builder notation:

It is also normal to show what type of number x is, like seen below:

It says "the set of all x's, such that x is greater than 0".

"the set of all x's that are a member of the Real Numbers, such that x is greater than or equal to 3"

   

• The • The

∈  means "a member of" or “an element of” (or simply "in"). is the special symbol for Real Numbers.

Number Types I showed you

Natural Numbers

(the special symbol for Real Numbers). Here are the common number types:

Integers

Rational Numbers

Real Numbers

   

       

Imaginary Numbers

Complex Numbers

Interval  Notation                

Correctly write the domain. The proper interval notation for the domain is easy to learn, but it is important that you write it correctly to express the correct answer and get full points on assignments and tests. Here are a few things you need to know about writing the domain of a function.

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The format for expressing the domain is an open bracket/parenthesis, followed by the 2 endpoints of the domain separated by a comma, followed by a closed bracket/parenthesis. For example, [-1,5). This means that the domain goes from -1 to 5.

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Use “[“ and “]” to indicate that a number is included in the domain. So in the example, [-1,5), the domain includes -1.

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Use “(“ and “)” to indicate that a number is not included in the domain. So in the example, [-1,5), 5 is not included in the domain. The domain stops arbitrarily short of 5, i.e. 4.999…

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Use “U” (meaning "union") to connect parts of the domain that are separated by a gap. For example, [-1,5) U (5,10]. This means that the domain goes from -1 to 10, inclusive, but that there is a gap in the domain at 5. This could be the result of, for example, a function with “x - 5” in the denominator.

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You can use as many "U" symbols as necessary if the domain has multiple gaps in it.

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Use infinity and negative infinity signs to express that the domain goes on infinitely in either direction.

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Always use ( ), not [ ], with infinity symbols.

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Examples To express that the domain of the function is all real numbers, write it like this: