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

Verified step by step guidance

1. <Step 1: Understand the definition of a function. A relation is a function if each input (or x-value) is associated with exactly one output (or y-value).>
2. <Step 2: Examine the given set of ordered pairs: \{(1,1), (1,-1), (0,0), (2,4), (2,-4)\}.>
3. <Step 3: Check for repeated x-values with different y-values. Notice that the x-value 1 is paired with both 1 and -1, and the x-value 2 is paired with both 4 and -4.>
4. <Step 4: Conclude that the relation does not define a function because there are x-values with more than one corresponding y-value.>
5. <Step 5: Identify the domain and range. The domain is the set of all x-values: \{1, 0, 2\}. The range is the set of all y-values: \{1, -1, 0, 4, -4\}.>

Key Concepts

Definition of a Function

A function is a specific type of relation where each input (or domain element) is associated with exactly one output (or range element). This means that for any given x-value, there cannot be multiple corresponding y-values. Understanding this definition is crucial for determining whether a relation qualifies as a function.

Domain and Range

The domain of a relation is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). Identifying the domain and range helps in understanding the behavior of the function and its limitations. For the given relation, one must extract these sets from the ordered pairs.

Ordered Pairs

Ordered pairs are pairs of numbers written in the form (x, y), where x is the first element and y is the second. In the context of relations, each ordered pair represents a connection between an input and an output. Analyzing these pairs is essential for determining if a relation is a function and for identifying the domain and range.