Example

Fill in the table.

Input	Output	Function?	Why or why not?
Name of senator	Name of state
Name of state	Name of senator
Time elapsed	Height of a tossed ball
Height of a tossed ball	Time elapsed
Number of cars	Number of tires
Number of tires	Number of cars

Answer:

Input	Output	Function?	Why or why not?
Name of senator	Name of state	Yes	For each input, there will only be one output because a senator only represents one state.
Name of state	Name of senator	No	For each state that is an input, 2 names of senators would result because each state has two senators.
Time elapsed	Height of a tossed ball	Yes	At a specific time, the ball has one specific height.
Height of a tossed ball	Time elapsed	No	Remember that the ball was tossed up and fell down. So for a given height, there could be two different times when the ball was at that height. The input height can result in more than one output.
Number of cars	Number of tires	Yes	For any input of a specific number of cars, there is one specific output representing the number of tires. (note: this would be assuming that all cars have four tires!)
Number of tires	Number of cars	Yes	For any input of a specific number of tires, there is one specific output representing the number of cars.

Example

List the domain and range for the following table of values where x is the input and y is the output.

x	y
[latex]−3[/latex]	[latex]4[/latex]
[latex]−2[/latex]	[latex]4[/latex]
[latex]−1[/latex]	[latex]4[/latex]
[latex]2[/latex]	[latex]4[/latex]
[latex]3[/latex]	[latex]4[/latex]

Answer: The domain describes all the inputs, and we can use set notation with brackets { } to make the list. [latex-display]\text{Domain}:\{-3,-2,-1,2,3\}[/latex-display] The range describes all the outputs. [latex-display]\text{Range}:\{4\}[/latex-display] We only listed [latex]4[/latex] once because it is not necessary to list it every time it appears in the range.

Example

Define the domain and range for the following set of ordered pairs, and determine whether the relation given is a function.

[latex]\{(−3,−6),(−2,−1),(1,0),(1,5),(2,0)\}[/latex]

Answer: We list all of the input values as the domain.  The input values are represented first in the ordered pair as a matter of convention. Domain: {[latex]-3,-2,1,2[/latex]} Note how we did not enter repeated values more than once; it is not necessary. The range is the list of outputs for the relation; they are entered second in the ordered pair. Range: {[latex]-6, -1, 0, 5[/latex]} Organizing the ordered pairs in a table can help you tell whether this relation is a function.  By definition, the inputs in a function have only one output.

x	y
[latex]−3[/latex]	[latex]−6[/latex]
[latex]−2[/latex]	[latex]−1[/latex]
[latex]1[/latex]	[latex]0[/latex]
[latex]1[/latex]	[latex]5[/latex]
[latex]2[/latex]	[latex]0[/latex]

The relation is not a function because the input [latex]1[/latex] has two outputs: [latex]0[/latex] and [latex]5[/latex].

Example

Find the domain and range of the relation and determine whether it is a function.

[latex]\{(−3, 4),(−2, 4),( −1, 4),(2, 4),(3, 4)\}[/latex]

Answer: Domain: {[latex]-3, -2, -1, 2, 3[/latex]} Range: {[latex]4[/latex]} To help you determine whether this is a function, you could reorganize the information by creating a table.

x	y
[latex]−3[/latex]	[latex]4[/latex]
[latex]−2[/latex]	[latex]4[/latex]
[latex]−1[/latex]	[latex]4[/latex]
[latex]2[/latex]	[latex]4[/latex]
[latex]3[/latex]	[latex]4[/latex]

Each input has only one output, and the fact that it is the same output (4) does not matter. This relation is a function.