Problem 62 - Algebra 2

Question

State the domain and range of each relation. Then graph the relation and determine whether it is a function. \(\\{(0,2),(1,3),(2,-1),(1,0)\\}\)

Step-by-Step Solution

Answer

Domain: \(\{0, 1, 2\}\); Range: \(\{2, 3, -1, 0\}\); Not a function.

Step 1: Identify the Domain

The domain is the set of all first elements (x-values) in the given relations. For the set \( \{(0,2),(1,3),(2,-1),(1,0)\} \), the domain is \( \{0, 1, 2\} \). Note that 1 is repeated, so we list it only once.

Step 2: Identify the Range

The range is the set of all second elements (y-values) in the given relations. For the set \( \{(0,2),(1,3),(2,-1),(1,0)\} \), the range is \( \{2, 3, -1, 0\} \).

Step 3: Graph the Relation

Plot each ordered pair on a coordinate plane. The pairs are \((0,2), (1,3), (2,-1), \text{and} (1,0)\).

Step 4: Determine if the Relation is a Function

A relation is a function if each x-value corresponds to only one y-value. In the set \( \{(0,2),(1,3),(2,-1),(1,0)\} \), the x-value 1 corresponds to both y-values 3 and 0. Therefore, it is not a function.

Key Concepts

Domain

In algebra, the domain of a relation refers to the set of all possible x-values (or inputs) that can be found within a given set of ordered pairs. It includes every first element in each of those pairs. Let's take the relation \[ \{(0,2),(1,3),(2,-1),(1,0)\} \] as an example. Here, the domain is the collection of x-values from these pairs: 0, 1, and 2. Even though the number 1 is listed twice, it's only counted once in the domain. So, the domain for this relation is \[ \{0, 1, 2\} \].

• The domain helps us to know which x-values are part of our relation.
• It's important because it defines the input values that we can have.

Understanding the domain is crucial in working with functions and graphs.

Range

The range is the counterpart to the domain. It includes all the y-values (or outputs) from the ordered pairs in a relation. For our example \[ \{(0,2),(1,3),(2,-1),(1,0)\} \], we look at the second element of each pair to determine the range. Thus, the range for this relation consists of the numbers 2, 3, -1, and 0.

• Unlike domain, here we gather all y-values from the pairs without repetition unless different x-values share the same y-value.
• Knowing the range helps understand what possible outputs we get from our inputs, whether we are dealing with equations, graphs, or functions.

Getting a good grasp of the range helps students predict behaviors of mathematical models.

Graphing Relations

When graphing a relation, our task is to take each ordered pair and plot it on a coordinate plane. Let's look at the pairs from the set \[ \{(0,2),(1,3),(2,-1),(1,0)\} \]. To graph this relation:

• Start by placing a point at \( (0, 2) \), where x is 0 and y is 2.
• Next, plot \( (1, 3) \), \( (2, -1) \), and \( (1, 0) \).

Each of these points corresponds to one of the ordered pairs in the relation.
By plotting them on the coordinate grid, we build a visual representation, which can help us identify patterns and particular traits of the relation. Graphs make concepts tangible and provide visual insight.

Function Determination

A relation is determined to be a function if each x-value in the domain corresponds to exactly one y-value. For the same set of ordered pairs \[ \{(0,2),(1,3),(2,-1),(1,0)\} \], observe that the x-value 1 maps to two different y-values (3 and 0). This means our relation is not a function.

• Functions must pass the "vertical line test", where no vertical line drawn would intersect the graph at more than one point.
• In simpler terms, if an x-value leads to more than one y-value, it's not a function.

Understanding whether a relation is a function is vital in higher mathematical reasoning and problem-solving. Functions form a fundamental concept in algebra and other advanced mathematical studies.