Domain and Range: Learn It 1

  • Find the domain of a function by looking at its equation
  • Find the domain of a function by looking at its graph
  • Sketch graphs of piecewise functions

Domain and Range

Now that we understand what functions are, let’s talk more about two important concepts related to them: the domain and the range.

Domain and Range

Domain: The domain of a function is the set of all possible input values. These are the values that you can put into the function.

  • Note that values in the domain are also known as input values, or values of the independent variable, and are often labeled with the lowercase letter [latex]x[/latex].

 

Range: The range of a function is the set of all possible output values. These are the values that come out of the function.

  • Values in the range are also known as output values, or values of the dependent variable, and are often labeled with the lowercase letter [latex]y[/latex].

We can write the domain and range in interval notation, which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [latex][[/latex] when the set includes the endpoint and a parenthesis [latex]([/latex] to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has [latex]$100[/latex] to spend, he or she would need to express the interval that is more than [latex]0[/latex] and less than or equal to [latex]100[/latex] and write [latex]\left(0,\text{ }100\right][/latex].

Before we begin, let us review the conventions of interval notation:

  • The smallest term from the interval is written first.
  • The largest term in the interval is written second, following a comma.
  • Parentheses, ( or ), are used to signify that an endpoint is not included, called exclusive.
  • Brackets, [ or ], are used to indicate that an endpoint is included, called inclusive.

Understanding the domain and range helps us to see the full scope of a function and how it operates over different values.

Consider the relation where the input is a family member’s name and the output is their age:

Family Member’s Name (Input) Family Member’s Age (Output)
Nellie [latex]13[/latex]
Marcos [latex]11[/latex]
Esther [latex]46[/latex]
Samuel [latex]47[/latex]
Nina [latex]47[/latex]
Paul [latex]47[/latex]
Katrina [latex]21[/latex]
Andrew [latex]16[/latex]
Maria [latex]13[/latex]
Ana [latex]81[/latex]

Domain: The domain is the set of all family members’ names:

Range: The range is the set of all family members’ ages: [latex]\{13,11,46,47,21,16,81\}[/latex]

Relations can be written as ordered pairs of numbers [latex](x,y)[/latex] or as numbers in a table of values the columns of which each contain inputs or outputs. By examining the inputs ([latex]x[/latex]-coordinates) and outputs ([latex]y[/latex]-coordinates), you can determine whether or not the relation is a function. Remember, in a function, each input corresponds to only one output. That is, each [latex]x[/latex] value corresponds to exactly one [latex]y[/latex] value.

Find the domain of the following function:

[latex]\left\{\left(2,\text{ }10\right),\left(3,\text{ }10\right),\left(4,\text{ }20\right),\left(5,\text{ }30\right),\left(6,\text{ }40\right)\right\}[/latex]

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