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It is common to want to make a connection between two different sets of information. For example, consider counting people at a park. The number of people changes as the temperatures changes. Such relations are used to describe a connection between the elements of two sets. This lesson will dive deep into relations and how to represent them.
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Tadeo, the younger brother of Vincenzo, loves watching Vincenzo's college basketball games. He recorded Vincenzo and his teammates' names, heights, and the number of $3-$point shots they made. Tadeo describes the relation between Vincenzo's teammates' height and the number of $3-$points shots they made. He is so excited to share them with his big sibling.

Which of the following diagrams can Tadeo use to illustrate the relation?{"type":"multichoice","form":{"alts":["Diagram I","Diagram II","Diagram III","Diagram IV"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":[0,1,2,3]}

A relation, or **relationship**, is a rule that relates the elements of one set to the elements of another set. The first set is called the set of inputs and the second set is called the set of outputs.

A mapping diagram is a graphic tool that helps to visualize a relation. In a mapping diagram, the inputs are listed in one set and the outputs in another. Then, arrows are drawn from each input to its corresponding outputs.

This mapping diagram shows the relation defined by the set ${(-2,-8),$ $(-1,11),$ $(0,5),$ $(2,-8)}.$ Note that $-8$ is the output of two different inputs, $-2$ and $2.$Tadeo enjoyed sharing his last finding so much that he now wants to examine the relationship between the height and age of Vincenzo's teammates. He wants to describe some relations based on the values in the following diagram.

Represent the relation between Vincenzo's teammates' ages and heights using a table of values, a mapping diagram, a set of ordered pairs, and points in a coordinate plane.**Table of Values:**

Ages | $16$ | $18$ | $21$ | $22$ | $21$ |
---|---|---|---|---|---|

Heights (m) | $1.70$ | $1.75$ | $1.85$ | $1.96$ | $1.91$ |

**Mapping Diagram:**

**Ordered Pairs:** ${(16,1.70),$ $(18,1.75),$ $(21,1.85),$ $(22,1.96),$ $(21,1.91)}$

**Coordinate Plane:**

To make a mapping diagram, place all the ages in one set and all the heights in another set. Then, connect each age with its corresponding height using an arrow. To write the relation as a set of ordered pairs, place the ages in the first component and the corresponding heights in the second component.

Using the information that Tadeo gathered, a relation between the ages and heights can be made and represented using different visualizations.

First, organize the relation using a table of values. Place the ages in the first row and the heights in the second row. Align the ages with the corresponding heights.

Ages | $16$ | $18$ | $21$ | $22$ | $21$ |
---|---|---|---|---|---|

Heights (m) | $1.70$ | $1.75$ | $1.85$ | $1.96$ | $1.91$ |

${(16,1.70),(18,1.75),(21,1.85),(22,1.96),(21,1.91) } $

Note that by hovering over each point, the corresponding information is shown.

The domain of a relation is the set of all inputs, for which the relation is defined. For example, consider the following relation.

Depending on how a relation is represented, its domain can be determined using different methods.

Depending on how a relation is represented, its range can be determined using different methods.

$R={(0,-2),(1,0),(2,2)} $

The domain of the relation is the set of all first coordinate of the ordered pairs. Relation | Inputs | Domain |
---|---|---|

${(0,-2),(1,0),(2,2)}$ | $0,$ $1,$ $2$ | ${0,1,2}$ |

The range of a relation is the set of all outputs of the relation. Consider the relation $R$ one more time.

$R={(0,-2),(1,0),(2,2)} $

By looking at the second coordinate of the ordered pairs, the range can be determined. Relation | Outputs | Range |
---|---|---|

${(0,-2),(1,0),(2,2)}$ | $-2,$ $0,$ $2$ | ${-2,0,2}$ |

If two different inputs have the same output, it is not necessary to repeat such output when writing the range.

Tadeo enjoyed displaying the relations between the ages and heights of Vincenzo's teammates so much that he wanted to go a step further and continue analyzing relations. Tadeo kept track of the scores in the first six games and Vincenzo's contribution to his team's score. What is the range of Relation $R?$
What is the range of Relation $F?$ ### Hint

### Solution

*only once*, the domain and range of $R$ are as follows.

In Relation $R,$ the $x-$values represent the score of Vincenzo's team, while the $y-$values indicate the opposing team's score. Relation $F$ shows Vincenzo's individual contribution to the score in each game. For instance, $(1,11)$ means that Vincenzo scores $11$ points in the first match.

a What is the domain of Relation $R?$

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{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":false,"numinput":6,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9189453125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Range<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">{<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">}<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["63","81","82","88","89","95"]}}

b What is the domain of Relation $F?$

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a The domain of a relation represented by a table is formed by the $x-$values, while its range is formed by the $y-$values. When writing sets, do not repeat elements.

b To find the domain, look at the set of inputs. Similarly, to find the range, look at the set of outputs.

a Given a relation represented by a table, the domain includes all the $x-$values, and the range includes all the $y-$values.

By listing the numbers from least to greatest, and including each element

$Domain ofRRange ofR ={67,77,86,93,95}={63,81,82,88,89,95} $

b As done in Part A, to find the domain of $F,$ look at the numbers inside the inputs set. To find the range, look at the outputs sets.

By listing the numbers from least to greatest, and including each element only once, the domain and range of $F$ are as follows.

$Domain ofFRange ofF ={1,2,3,4,5,6}={11,17,20,27,29} $

Tadeo is on a roll with his basketball analyses. Other than the game itself, he has become fascinated with the crowd.
He writes a relation showing the number of spectators in each game. In addition to that, he then writes another relation that shows the revenue from the tickets sold to these rambunctious fans. {"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":false,"numinput":6,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Domain<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">{<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">}<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["1","2","3","4","5","6"]}} What is the range of Relation $P?$
What is the range of Relation $Q?$ ### Hint

b To find the domain, look at the set of inputs. Similarly, to find the range, look at the set of outputs.
### Solution

*only once*, the domain and range of $P$ are as follows.

The $x-$coordinates of the ordered pairs in Relation $P$ represent the number of the match, while the $y-$coordinates represent the number of spectators in the match. Relation $Q$ shows the number of spectators on the $x-$axis and revenue in dollars on the $y-$axis.

a What is the domain of Relation $P?$

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b What is the domain of Relation $Q?$

{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":false,"numinput":6,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.73046875em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Domain<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">{<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">}<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["100","110","120","125","135","145"]}}

{"type":"text","form":{"type":"list","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]},"ordermatters":false,"numinput":6,"listEditable":true,"hideNoSolution":true},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9189453125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Range<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">{<\/span><\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.80002em;vertical-align:-0.65002em;\"><\/span><span class=\"mord\"><span class=\"delimsizing size2\">}<\/span><\/span><\/span><\/span><\/span>","answer":{"text":["500","550","600","625","675","725"]}}

a The domain of a relation represented by a set of ordered pairs is formed by the first coordinate of each pair, while its range is formed by the second coordinates. When writing sets, do not repeat elements.

a Given a set of coordinate pairs, the domain includes all the first coordinates, and the range includes all the second coordinates.

By listing the numbers from least to greatest, and including each element

$Domain ofPRange ofP ={1,2,3,4,5,6}={100,110,120,125,135,145} $

b To find the domain of $Q,$ look at the $x-$coordinates of the points. To find the range, look at the $y-$coordinates of the points.

By listing the numbers from least to greatest, and including each element only once, the domain and range of $Q$ are as follows.

$Domain ofQRange ofQ ={100,110,120,125,135,145}={500,550,600,625,675,725} $

The following applet displays a relation either as a table, a set of ordered pairs, a mapping diagram, or a group of coordinate points in the coordinate plane. For any given relation, determine its domain and range.

The following applet displays two relations. Determine if they represent the same relation.

Consider all of the diagrams given at the beginning of the lesson. Compare the information that each diagram provides and see if they correspond to each other.

The first diagram shows a relation represented as a table. It consists of rows and columns, where each row represents an ordered pair. The first column represents the input values ($x-$values), and the second column represents the output values ($y-$values).

To compare with other representations, ensure that the values in the table match the input-output pairs provided in other diagrams.

The second diagram visually represents a relation by using arrows to connect the input values with their corresponding output values. The input values are shown on the left side, and the output values on the right side.

Notice that the arrows in the mapping diagram correctly link the input values to their respective output values. This is the same as the pairs shown in the table.

${(1.70,2),(1.75,5),(1.85,4),(1.91,8),(1.96,8) } $

This is also the same as the ordered pairs shown in the previous diagrams. Finally, the fourth diagram shows a relation as a group of coordinate points in the coordinate plane. Recall that each point consists of an $x-$coordinate and a $y-$coordinate.

The points on the coordinate plane match with the input-output pairs provided in other representations. As a result the information matches across all representations. It can be concluded that they represent the same relation.