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Chapter 2: Problem 43

Determine the domain and range of each relation. $$\\{(-3,1),(4,2),(-3,6),(5,6)\\}$$

Short Answer

Expert verified
Domain: \( \{-3, 4, 5\} \). Range: \( \{1, 2, 6\} \).

Step by step solution

01

- Identify all the x-values

The x-values in the set of ordered pairs are the first elements of each pair. Extract the x-values from each pair: \((-3,1)\rightarrow x=-3, (4,2)\rightarrow x=4, (-3,6)\rightarrow x=-3, (5,6)\rightarrow x=5\).
02

- Determine the domain

The domain includes all unique x-values identified. Here, we have \( x = -3, 4, 5 \). So, the domain is \( \{-3, 4, 5\} \).
03

- Identify all the y-values

The y-values in the set of ordered pairs are the second elements of each pair. Extract the y-values from each pair: \((-3,1)\rightarrow y=1, (4,2)\rightarrow y=2, (-3,6)\rightarrow y=6, (5,6)\rightarrow y=6\).
04

- Determine the range

The range includes all unique y-values identified. Here, we have \( y = 1, 2, 6 \). So, the range is \( \{1, 2, 6\} \).

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Key Concepts

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

unique x-values
To determine the domain of a relation, we need to focus on the unique x-values. The x-values in our ordered pairs are the first elements from each pair. In this relation \{(-3,1), (4,2), (-3,6), (5,6)\}, we extract the x-values as follows: \((-3, 1)\rightarrow x=-3\), \((4, 2)\rightarrow x=4\), \((-3, 6)\rightarrow x=-3\), and \((5, 6)\rightarrow x=5\). We list them but ensure not to repeat any values. Hence, the unique x-values here are \(-3, 4, 5\). It's crucial to only include each x-value once in our domain. Thus, the domain for this relation is \{-3, 4, 5\}.
identifying y-values
Next, we need to identify the range of the relation by looking at the y-values. The y-values are the second elements in the ordered pairs. For our set of pairs \{(-3,1), (4,2), (-3,6), (5,6)\}, we extract the y-values from each: \((-3, 1)\rightarrow y=1\), \((4, 2)\rightarrow y=2\), \((-3, 6)\rightarrow y=6\), and \((5, 6)\rightarrow y=6\). Similar to the domain, we need to list these values but ensure to avoid any duplicates. The unique y-values here are \{1, 2, 6\}. Thus, the range for this relation is \{1, 2, 6\}.
ordered pairs
In mathematics, an ordered pair is a pair of elements in a specific order, typically written as \(x, y\). For any given relation, the domain consists of all unique x-values, and the range consists of all unique y-values. The ordered pairs in our exercise are \{(-3, 1), (4, 2), (-3, 6), (5, 6)\}. Each pair connects an x-value with a y-value, indicating a relationship between them. Recognizing this relationship helps in understanding the overall behavior of the set. For example, the ordered pair \(4, 2\)\ tells us that when x is 4, y is 2. Understanding and analyzing ordered pairs is crucial for grasping more complex mathematical concepts such as functions and graphs.

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Most popular questions from this chapter

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