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Chapter 3: Problem 7

For exercises 7-20, identify the set as a relation, a function, or both a relation and a function. $$ \\{(5,4),(9,2),(13,0)\\} $$

Short Answer

Expert verified
Both a relation and a function.

Step by step solution

01

Define Relation

A relation is any set of ordered pairs. Check the given set: \(\{(5,4),(9,2),(13,0)\}\). Since it is a set of ordered pairs, it qualifies as a relation.
02

Define Function

A function is a special type of relation where each input (or first element of the ordered pair) maps to exactly one output (or second element).
03

Check for Unique Inputs

To check if the given set is a function, ensure that each input value is unique. The inputs in the set are 5, 9, and 13. All these inputs are unique.
04

Conclusion

Since each input maps to exactly one output and the inputs are unique, the given relation \(\{(5,4),(9,2),(13,0)\}\) is both a relation and a function.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Relation Definition
A relation is a set that contains ordered pairs. In simpler terms, it links or relates certain values together. An ordered pair consists of two elements: the first element (which we call the input) and the second element (which we call the output). For example, in the ordered pair \(5, 4\), 5 is the input and 4 is the output.
To identify a relation, you just need to see if you have a collection of these ordered pairs. It doesn鈥檛 matter whether the inputs or outputs repeat themselves.
In our given set \(\{(5, 4), (9, 2), (13, 0)\}\), we have a collection of three ordered pairs. Therefore, this set is a relation.
Function Definition
A function is a special type of relation. Every function is a relation, but not every relation is a function. What makes a function unique is that each input, or first element, maps to exactly one output, or second element. In other words, for a set to be a function, no input value should repeat with different output values.
Let's say you have the ordered pairs \(3, 8\) and \(3, 2\). Since the input '3' is associated with two different outputs (8 and 2), this set would not be a function.
Unique Input Values
To determine if a relation is also a function, you must check whether each input value in the set is unique. In our set \(\{(5, 4), (9, 2), (13, 0)\}\), the input values are 5, 9, and 13.
Since all these inputs are unique and none of them are repeated, this set of ordered pairs is indeed a function.
So, to sum it up:
  • If you have a set of ordered pairs where no input is duplicated, then you have a function.
  • If any input in the ordered pairs is duplicated with different output values, it is just a relation, not a function.
      • Saying this, our set \(\{(5, 4), (9, 2), (13, 0)\}\) is both a relation and a function because it passes the unique input test.

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Most popular questions from this chapter

For exercises 97-98, some students find it helpful to use their learning preferences as a guide in how to study. Visual Learner \- Take detailed notes during class. Use colored pens and highlighters. \- Reorganize and rewrite notes after class; draw diagrams that summarize what you have learned. \- Read your book; watch the videos or DVDs for this text. \- Use flash cards for memory work. \- Sit where you can see everything in the classroom. Turn your phone or tablet off so that you are not distracted. Auditory Learner \- With permission, record your class. Take only brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Restate the main ideas aloud to yourself. Use videos and DVDs to fill in anything you missed in class. \- Talk to yourself as you do your homework. Explain each step to yourself. \- Do memory work by repeating definitions aloud. Listen to a recording of the words and definitions. Create songs that help you remember a definition. \- Sit where you can hear everything. Turn your phone or tablet off so that you are not distracted. Kinesthetic Learner \- With permission, record your class. Take brief notes of the big ideas and examples. After class, listen to the recording. Complete your notes. Draw pictures. Use videos and DVDs to fill in anything you missed during class. -With your finger, trace diagrams and graphs. Do not just look at them. \- Imagine symbols such as variables as three-dimensional objects or even cartoon characters. Imagine yourself counting them, combining them, or subtracting them. \- Do memory work as you exercise or walk to your car. Walk around your room as you repeat definitions. You may find it helpful to come up with physical motions and/or a song that correspond to a procedure. \- If your class is mostly lecture, prepare yourself mentally before you walk into class to concentrate and not daydream. Turn your phone or tablet off so that you are not distracted. Identify any of the strategies listed above that you currently use to study math.

Use the slope formula to find the slope of the line that passes through the points. \((5,10) ;(7,16)\)

(a) represent the information as two ordered pairs. (b) find the average rate of change, \(m\). The number of traffic fatalities in Missouri decreased from 1257 deaths in 2005 to 819 deaths in 2010. Round to the nearest whole number. (Source: www- nrd.nhtsa .dot.gov)

(a) find the \(y\)-intercept. (b) find the \(x\)-intercept. (c) use the slope formula to find the slope of the line. \(-8 x+3 y=48\)

(a) write the equation of the horizontal line that passes through the point. (b) graph the equation. \((-3,-2)\)
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