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

Verified step by step guidance

1. Recall that a relation defines a function if every input (x-value) corresponds to exactly one output (y-value).
2. Examine the given set of ordered pairs: \(\{(1,1), (1,-1), (0,0), (2,4), (2,-4)\}\). Identify the x-values: 1, 1, 0, 2, 2.
3. Notice that the x-values 1 and 2 each appear more than once but are paired with different y-values (1 and -1 for x=1; 4 and -4 for x=2). This means the relation does not assign a unique output for these inputs.
4. Conclude that this relation does not define a function because some inputs have multiple outputs.
5. To find the domain, list all unique x-values: \(\{0, 1, 2\}\). To find the range, list all unique y-values: \(\{1, -1, 0, 4, -4\}\).

Key Concepts

Definition of a Function

A function is a relation where each input (or domain element) corresponds to exactly one output (or range element). If any input is paired with more than one output, the relation is not a function. This concept helps determine if the given set of ordered pairs qualifies as a function.

Domain of a Relation

The domain is the set of all possible input values (first elements) in the relation. Identifying the domain involves listing all unique x-values from the ordered pairs. Understanding the domain is essential for describing the inputs over which the relation or function is defined.

Range of a Relation

The range is the set of all possible output values (second elements) in the relation. It includes all unique y-values from the ordered pairs. Knowing the range helps describe the outputs that the relation or function can produce.