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Chapter 3: Problem 27

Find the domain and range of each relation. Then determine whether the relation represents a function. \\{(3,3),(3,5),(0,1),(-4,6)\\}

Short Answer

Expert verified
Domain: \( \{3, 0, -4\} \). Range: \( \{3, 5, 1, 6\} \). Not a function.

Step by step solution

01

- Identify the Domain

The domain of a relation is the set of all possible input values (x-coordinates). For the given set of ordered pairs \{(3,3),(3,5),(0,1),(-4,6)\}, the x-coordinates are 3, 3, 0, and -4. The domain is \( \{3, 0, -4\} \).
02

- Identify the Range

The range of a relation is the set of all possible output values (y-coordinates). For the given set of ordered pairs \{(3,3),(3,5),(0,1),(-4,6)\}, the y-coordinates are 3, 5, 1, and 6. The range is \( \{3, 5, 1, 6\} \).
03

- Determine if the Relation is a Function

A relation is a function if each input value (x-coordinate) is associated with exactly one output value (y-coordinate). In the given relation, the x-coordinate 3 is associated with two different y-coordinates (3 and 5), so this relation is not a function.

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Key Concepts

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

Relation
In mathematics, a relation is a set of ordered pairs \((x,y)\). Each ordered pair consists of an input value (x-coordinate) and an output value (y-coordinate). Understanding relations is crucial because they help us see how different elements from one set relate to elements of another set. Relations can be visualized as lists of pairs, mappings, or even graphs.
Function
A function is a special type of relation where each input value (x-coordinate) is paired with exactly one output value (y-coordinate). This means no two different pairs can have the same first element but different second elements. In other words, for a relation to be a function, every x-coordinate must map to a single y-coordinate. If any input value maps to more than one output value, the relation is not a function.
Algebra
Algebra involves working with numbers, variables, and operations to establish relationships and solve equations. Understanding the concept of domain and range is essential in algebra, as it helps delineate the scope of the relations and functions.

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Most popular questions from this chapter

Explain how the domain of \(g(x)=\sqrt{x}\) compares to the domain of \(g(x-k),\) where \(k \geq 0\).

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The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and Fahrenheit \(\left({ }^{\circ} \mathrm{F}\right)\) scales for measuring temperature is given by the equation $$ F=\frac{9}{5} C+32 $$ The relationship between the Celsius \(\left({ }^{\circ} \mathrm{C}\right)\) and \(\mathrm{Kelvin}(\mathrm{K})\) scales is \(K=C+273 .\) Graph the equation \(F=\frac{9}{5} C+32\) using degrees Fahrenheit on the \(y\) -axis and degrees Celsius on the \(x\) -axis. Use the techniques introduced in this section to obtain the graph showing the relationship between Kelvin and Fahrenheit temperatures.
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