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Chapter 9: Problem 25

Determine whether each relation defines a function, and give the domain and range. $$ \\{(1,1),(1,-1),(0,0),(2,4),(2,-4)\\} $$

Short Answer

Expert verified
The relation is not a function. The domain is {0, 1, 2}, and the range is {0, 1, -1, 4, -4}.

Step by step solution

01

- Understand the Concept of a Function

A relation is a function if every element in the domain is paired with exactly one element in the range. This means that no x-value (input) is associated with more than one y-value (output).
02

- Identify the Domain

Identify the unique x-values from the given set of pairs. The domain includes all distinct x-values: 1, 0, and 2.
03

- Check for Unique Outputs

Examine if every x-value is paired with exactly one y-value. Here, the pairs are 1 鈫� 1 and 1 鈫� -1, 2 鈫� 4 and 2 鈫� -4. Since x-values 1 and 2 have more than one corresponding y-value, the relation is not a function.
04

- Identify the Range

Identify the y-values from the given pairs. The range includes all distinct y-values: 1, -1, 0, 4, and -4.

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Key Concepts

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

Domain and Range
When working with functions and relations, understanding the domain and range is crucial. The domain of a relation is the set of all possible input values (usually represented by x-values). To find the domain, simply list all the unique x-values from the given pairs. In the exercise provided, the domain consists of the points where the x-values are 1, 0, and 2.

Next, the range is the set of all possible output values (usually represented by y-values). From the given pairs, we identify the y-values to determine the range. In this exercise, the range includes y-values 1, -1, 0, 4, and -4. By knowing the domain and range, you can get a clearer picture of how the relation behaves, laying the foundation for further analysis of whether it qualifies as a function.
Unique Outputs
A key characteristic of a function is that each input (x-value) must map to exactly one output (y-value). This consistency ensures that for any value in the domain, there is a distinct and unique corresponding value in the range. This concept distinguishes functions from other kinds of relations.

Take a closer look at the given pairs: (1,1), (1,-1), (0,0), (2,4), and (2,-4). Notice how the x-value 1 maps to both 1 and -1, and the x-value 2 maps to both 4 and -4. These multiple mappings for a single x-value indicate that this relation is not a function. In a valid function, each x-value should be associated with only one y-value.
Function Definition
Understanding what defines a function is fundamental in various fields of mathematics. A function is a type of relation where each element from the domain is paired with exactly one element from the range. In other words, it must have a unique output for each input.

Given the relation from the exercise, it doesn't satisfy this definition due to the fact that certain x-values (1 and 2 in this case) are linked with more than one y-value. It's important to identify such discrepancies to determine if a relation qualifies as a function. Understanding this property helps in performing further mathematical operations and analyses correctly.

To summarize: if any x-value points to more than one y-value, the relation is not defined as a function.

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