Problem 53 - Algebra 1: Concepts and Skills

Question

Determine whether the relation is a function. If it is a function, give the domain and range. $$ (-2,-2),(0,0),(1,1),(2,2) $$

Step-by-Step Solution

Answer

The relation is a function. The domain of the function is {-2, 0, 1, 2}, and the range is also {-2, 0, 1, 2}.

Step 1: Identify if the relation is a function

Begin by examining each given ordered pair in the relation. Checking the ordered pairs (-2,-2), (0,0), (1,1), (2,2), each x-value (input) maps to exactly one y-value (output). Therefore, the given relation is a function.

Step 2: Determine the domain

The domain of a function consists of all the possible x-values. Here, the domain is the set of all x-values in the given ordered pairs. Looking at the ordered pairs, the x-values are -2, 0, 1, and 2.

Step 3: Determine the range

The range of a function consists of all the possible y-values. For this function, the y-values in the ordered pairs are -2, 0, 1, and 2.

Key Concepts

Relation

A **relation** in mathematics refers to a set of ordered pairs, where each pair consists of two elements. The first element is typically called the input or the x-value, while the second element is referred to as the output or the y-value. Relations establish a connection between these pairs, indicating how elements of one set relate to elements of another.

• In the given exercise, the relation is a set of ordered pairs: \((-2,-2), (0,0), (1,1), (2,2)\).
• Relations can be visualized in a table or a graph, showing the connections between x-values and y-values.

To determine if a relation is a function, check if each x-value pairs with exactly one y-value. If every x-value is unique to one y-value, the relation is a function, as confirmed in this exercise. This is an important characteristic because functions have specific rules assigning outputs to inputs consistently.

Domain

The **domain** of a function is the collection of all possible input values (x-values) that can be used in the function. To find the domain, you simply extract all the x-values from each ordered pair in the relation.

• For the exercise at hand, the ordered pairs are \((-2,-2), (0,0), (1,1), (2,2)\).
• The x-values are {-2, 0, 1, 2}.

Listing these values, we define the domain of the function as \(\{-2, 0, 1, 2\}\). These values inform us about all the x-values that the function will accept. Knowing the domain helps us understand the scope of input values and analyze the behavior of the function over these inputs.

Range

The **range** of a function consists of all the potential outputs or y-values that the function produces when we use each value in the domain. Similar to how we find the domain, we can find the range by looking at the y-values in the set of ordered pairs.

• For the given relation, the ordered pairs are \((-2,-2), (0,0), (1,1), (2,2)\).
• The y-values are {-2, 0, 1, 2}.

Thus, the range is \(\{-2, 0, 1, 2\}\). It shows us which values can be produced by the function based on the inputs. Understanding the range ensures that we know all potential y-values that the function can achieve and allows us to visualize the full set of outputs.