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Chapter 7: Problem 9

Decide whether each relation defines a function, and give the domain and range. See Examples \(I-4\) $$ \\{(5,1),(3,2),(4,9),(7,6)\\} $$

Short Answer

Expert verified
The relation is a function. Domain: \{5, 3, 4, 7\}. Range: \{1, 2, 9, 6\}.

Step by step solution

01

- Understand Definition of Function

A relation is a function if every input (or domain element) has exactly one output (or range element). In other words, each x-value must be associated with only one y-value.
02

- Identify Domain and Range

Identify the set of all x-values (domain) and the set of all y-values (range) in the given relation. Here, the pairs are (5, 1), (3, 2), (4, 9), and (7, 6).
03

- List the Domain

The domain are the x-values from each pair: \[ \{5, 3, 4, 7\} \]
04

- List the Range

The range are the y-values from each pair: \[ \{1, 2, 9, 6\} \]
05

- Check Uniqueness of Domain Values

Ensure that each x-value in the domain maps to exactly one y-value. Since each x-value 5, 3, 4, and 7 maps to 1, 2, 9, and 6 respectively without repetition, the relation is a function.

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Key Concepts

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

Relation
In algebra, a relation is simply a set of ordered pairs. Each pair consists of an input value (usually called x) and an output value (usually called y). In simpler terms, a relation connects inputs to outputs.
A key point to understand is that a relation does not need to follow the rules as stringently as a function. For example, one input can connect to multiple outputs. This is common in broader forms of data collections that don't define functions.
To illustrate, consider the relation described in the exercise: \( \{(5,1),(3,2),(4,9),(7,6)\} \). Each pair in this set can be described as one x-value paired with one y-value.
Domain and Range
The domain and range are fundamental concepts when discussing functions and relations.
Domain: The domain is the set of all possible input values (x-values) in the relation. For our given set of pairs, the domain consists of: \[ \{5, 3, 4, 7\} \].
Range: The range is the set of all possible output values (y-values) in the relation. For our given set, the range consists of: \[ \{1, 2, 9, 6\} \].
These sets include all the x and y values found in the pairs, giving a complete picture of what inputs map to what outputs.
Uniqueness of Domain Values
For a relation to be classified as a function, each input value (x-value) must map to exactly one output value (y-value). This means no x-value should be paired with more than one y-value.
In our exercise, each x-value from the set \( \{5, 3, 4, 7\} \) maps to a unique y-value from the set \( \{1, 2, 9, 6\} \), without any repetition or overlap. Specifically:
  • 5 maps to 1
  • 3 maps to 2
  • 4 maps to 9
  • 7 maps to 6
This confirms that the given relation is indeed a function, as each domain value has a unique range value.

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