Problem 30 - A Graphical Approach to College Algebra

Question

Determine the domain \(D\) and range \(R\) of each relation, and tell whether the relation is a function. Assume that a calculator graph extends indefinitely and a table includes only the points shown. $$\\{(0,5),(1,3),(0,-4)\\}$$

Step-by-Step Solution

Answer

Domain: \(\{0, 1\}\), Range: \(\{5, 3, -4\}\), Not a function.

Step 1: Identify the Set of Ordered Pairs

The relation is given by the set of ordered pairs \( \{(0,5),(1,3),(0,-4)\} \). Each ordered pair consists of an input (first element) and an output (second element).

Step 2: Determine the Domain

The domain of a relation consists of all the first elements (or inputs) of the ordered pairs. For the given set \( \{(0,5),(1,3),(0,-4)\} \), the domain is \( \{0, 1\} \), because these are the distinct input values.

Step 3: Determine the Range

The range of a relation comprises all the second elements (or outputs) of the ordered pairs. For the set \( \{(0,5),(1,3),(0,-4)\} \), the range is \( \{5, 3, -4\} \), representing all the distinct output values.

Step 4: Assess If the Relation is a Function

To determine if the relation is a function, each input should have exactly one output. In the set \( \{(0,5),(1,3),(0,-4)\} \), the input \(0\) is associated with two different outputs (\(5\) and \(-4\)), so the relation is not a function.

Key Concepts

Ordered Pairs

Ordered pairs are fundamental in understanding relations between sets of numbers. Each ordered pair consists of two elements: an input (the first number) and an output (the second number). For example, in the ordered pair \((0,5)\), the input is 0, and the output is 5. These pairs are a formal way to represent relationships where each input is associated with precisely one output.

When dealing with multiple ordered pairs, it's vital to differentiate between distinct inputs and outputs. The collection of inputs forms what we call the "domain," while the collection of outputs makes up the "range." Identifying ordered pairs correctly helps us understand how inputs and outputs interact in mathematical and real-world contexts. Recognizing this structure is crucial as it underpins more advanced concepts like functions and graphs.

Ordered pairs can be visualized on a coordinate plane, where each pair takes on a position specified by its input and output. This visualization helps to emphasize how different inputs relate to their corresponding outputs, making it easier to spot patterns and relationships.

Functions

A function is a special kind of relation where each input corresponds to exactly one output. In other words, no input in a function can be associated with more than one output. Understanding functions is pivotal in mathematics because they model consistent and predictable systems.

To determine whether a set of ordered pairs is a function, check if any input repeats with different outputs. If every input maps to one unique output, then the relation is indeed a function.

Consider the ordered pairs \((0,5)\), \((1,3)\), and \((0,-4)\). The input 0 appears twice with differing outputs: 5 and -4. This violates the rule that each input in a function must have a single output, so these ordered pairs do not represent a function. Functions are ideal in scenarios where each input must lead to a consistent outcome, such as computing taxes or population growth modeling.

Graphical Representation

Graphically representing relations and functions is beneficial because it provides a visual summary of how inputs relate to outputs.
In a graph, ordered pairs are plotted as points on a coordinate plane. Each point has coordinates corresponding to the input and output in the relation. This makes it easy to visualize and analyze the structure of the relationship.
If a relation qualifies as a function, its graph has a specific characteristic: it will pass the "vertical line test." This test involves drawing vertical lines across the graph to ensure that each line intersects the graph at no more than one point. If any vertical line crosses the graph at more than one point, the graph is not a function. This is because multiple intersections indicate that a single input has been paired with multiple outputs.

• Graphs help identify whether a relation is a function quickly.
• Patterns within data become more apparent.
• Dynamic relationships are easier to interpret.

Graphical representations transform complex relationships into understandable visuals, making them a powerful tool both within academic settings and in practical applications.