Problem 29 - 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. $$\\{(4,1),(3,-5),(-2,3),(3,7)\\}$$

Step-by-Step Solution

Answer

The domain is \(\{4, 3, -2\}\), the range is \(\{1, -5, 3, 7\}\), and the relation is not a function.

Step 1: Identify the Domain

The domain of a relation is the set of all possible inputs (x-values). For the given set of points \(\{(4,1),(3,-5),(-2,3),(3,7)\}\), the x-values are 4, 3, -2, and 3. Although 3 appears twice, we only list it once. Therefore, the domain is \(\{4, 3, -2\}\).

Step 2: Identify the Range

The range of a relation is the set of all possible outputs (y-values). For the set \(\{(4,1),(3,-5),(-2,3),(3,7)\}\), the y-values are 1, -5, 3, and 7. Thus, the range is \(\{1, -5, 3, 7\}\).

Step 3: Determine Whether the Relation is a Function

A relation is a function if each input (or x-value) corresponds to exactly one output (or y-value). In the given set of points, the x-value 3 corresponds to two different y-values, -5 and 7. Therefore, this relation is not a function.

Key Concepts

Functions

In mathematics, a function is a fundamental concept that describes an input-output relationship. Imagine a vending machine: you select a product (input), and it delivers that specific item (output). For a relation to be considered a function, every input must relate to a single output. This means:

• Each element in the domain (list of all possible inputs) can correlate to only one element in the range (list of all possible outputs).
• Repeated inputs with different outputs, as seen in the set \(\{(4,1),(3,-5),(-2,3),(3,7)\}\), indicate the relation is not a function.

Understanding whether a relation is a function is crucial in many areas of mathematics and its applications. Functions are significant because they ensure a consistent and predictable behavior where each input generates exactly one result.

Graphical Representation

Graphical representation is simply a visual way to understand concepts like functions, relations, domain, and range. Imagine plotting points on a graph, such as those from our exercise: \((4,1), (3,-5), (-2,3), (3,7)\). Each coordinate is a point on the graph, letting us view where they lie in the space:

• It helps us identify patterns or behaviors of a relation.
• Graphically, a function typically passes the "vertical line test" where a vertical line interacts with the graph at no more than one point.

In our case, the x-value 3 appears twice with different y-values, so if plotted, a vertical line at \(x=3\) would intersect the graph at two different points, illustrating the relation isn’t a function. Graphical representations make it easier to visually verify domain, range, and functionality of a relation by simply observing relationships and intersections.

Input-Output Relationship

The input-output relationship describes the connection between two quantities: inputs which are the domain, and outputs which become the range. Think of this relationship as a black box. You put in an input, and something specific comes out as output. Each point in a set like \(\{(4,1),(3,-5),(-2,3),(3,7)\}\) signifies how inputs and outputs pair:

• Inputs are represented by x-values, highlighting what is possible to provide into the system.
• Outputs, or y-values, represent what the system returns.

This relationship is essential in validating whether a pattern is a function. For a function, each input is linked to a single output unique to it. When broken, as in our example where the input 3 provides multiple outputs, it reinforces that the connection does not form a function. Understanding this relationship is key in realizing how systems and real-world processes work consistently.