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Finding Domains and Ranges Using Potato Diagrams and Sets of Ordered Pairs or Tables

Precalculus Algebra Domains and Ranges of Functions Finding Domains and Ranges Using Potato Diagrams and Sets of Ordered Pairs or Tables

Domains and Ranges of Functions in Three Flavors

Recall…

The Domain of a function is the set of all possible inputs/$x$-values for which the function is defined (i.e. all possible inputs that the function can return an output for). The domain of a function $f$ is often denoted $\text{dom}(f)$.

The Range of a function is the set of all possible outputs/$y$-values the function produces from its inputs. The range of a function $f$ is often denoted $\text{ran(f)}$.

When it comes to finding the domain and range of a function defined by a table of values, potato diagram, or set of ordered pairs, the method is simple and essentially the same across all three types of representations.

Finding Domains and Ranges Using Potato Diagrams, Tables, and Sets of Ordered Pairs

To find the domain, look for and list all possible $x$-values that the function has a defined output for. Usually these values are in the left column of a table, the elements in the left potato in a potato diagram, and the $x$-values in a list of ordered pairs.

To find the range, look for and list all possible $y$-values that come out of the function as outputs. Usually these values are in the right column of a table, the elements in the right potato in a potato diagram, and the $y$-values in a list of ordered pairs.

See if you can complete some of the examples below using the above method for finding domains and ranges!

$x$	$f(x)=y$
1	-551
4	4
6	5412
7	-41
2	undefined
9	10

$dom(f)=\{1,4,6,7,9\}$ and $ran(f)=\{-551,4,5412, -41, 10\}$

For the domain, we are looking for the set of all $x$-values that the function (as defined by the table) has a defined output for. In this case, we have a defined output for every $x$-value except $2$, so $2$ is not in the domain of $f$.

For the range, we are looking for the set of all possible outputs, or $y$-values returned by the function. In this case, the $y$-values are the ones given in the list. We don’t list the word “undefined” because it indicates that there is no output that corresponds with the input of $2$.

$dom(f)=\{-5,3.14,12, 0,1\}$ and $ran(f)=\{6,0,-54\}$

For the domain, we are looking for all possible $x$-values that have a corresponding $y$-value in the list. In other words, list all possible $x$-values from the list of $(x,y)$ pairs in the given set.

For the range, we are looking for all possible output $y$-values that the function returns. In other words, we are looking for a list of all possible $y$-values from the list of $(x,y)$ pairs in the given set, without listing duplicates.

Important point: The number of elements in the range of a function can never exceed the number of elements in the domain. For, otherwise, the function would have some inputs going to multiple outputs, which is contrary to how functions work. Functions, by definition, take one input and return exactly one possible output… never more than one output. It IS possible for several different $x$-values to map to the same $y$-value, but you cannot have one $x$-value that produces more than one output.

dom$(f)=\{1,3,4,5\}$

ran$(f)=\{7,3,2\}$

For the domain, i.e. the set of all inputs that the function produces an output for, notice that the only numbers in the left bubble that have arrows coming out of them are 1,3,4, and 5. While 2 appears in the left potato, there is no arrow coming out of it, and therefore no output for an input of 2. Therefore, the domain is $\{1,3,4,5\}$.

For range, the only numbers in the right potato that have an arrow pointing to them are 7,3, and 2. So, those numbers are the only outputs of the function. Hence, the range is $\{7,3,2\}$

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