In Exercises 1–10, determine whether each relation is a function. Give the domain and range for each relation. {(1, 4), (1, 5), (1, 6)}

Verified step by step guidance

1. Step 1: Understand the definition of a function. A relation is a function if each input (domain value) is associated with exactly one output (range value). In other words, no input can have multiple outputs.
2. Step 2: Examine the given relation {(1, 4), (1, 5), (1, 6)}. Identify the inputs (domain values) and outputs (range values). Here, the input values are all '1', and the output values are '4', '5', and '6'.
3. Step 3: Check if the input values are unique or repeated. In this case, the input '1' is repeated and associated with multiple outputs ('4', '5', and '6'). This violates the definition of a function.
4. Step 4: Determine the domain and range of the relation. The domain is the set of all input values, which is {1}. The range is the set of all output values, which is {4, 5, 6}.
5. Step 5: Conclude whether the relation is a function. Since the input '1' is associated with multiple outputs, the relation is NOT a function. The domain is {1}, and the range is {4, 5, 6}.

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 no two ordered pairs can have the same first element with different second elements. Understanding this definition is crucial for determining if a given relation qualifies as a function.

Domain and Range

The domain of a relation is the set of all possible input values (first elements of the ordered pairs), while the range is the set of all possible output values (second elements). Identifying the domain and range helps in understanding the behavior of the relation and is essential for analyzing functions.

Ordered Pairs

Ordered pairs are pairs of numbers written in the form (x, y), where 'x' represents the input and 'y' represents the output. In the context of relations, analyzing the ordered pairs helps to determine if the relation meets the criteria of a function, particularly by checking for repeated first elements with different second elements.