Problem 82 - College Algebra with Modeling and Visualization

Question

Determine if \(S\) is a function. \(S\) is given by the table. $$ \begin{array}{llll} x & 1 & 2 & 3 \\ y & 1 & 1 & 1 \end{array} $$

Step-by-Step Solution

Answer

Yes, \(S\) is a function because each input has exactly one output.

Step 1: Understanding the definition of a function

A function is defined as a relationship between a set of inputs, called the domain, and a set of possible outputs, called the codomain, such that each input is related to exactly one output. This means every unique input value should map to one and only one output value.

Step 2: Identifying inputs and outputs

In the given table, the input values (domain) are the values in the first row: \(x = 1, 2, 3\). The corresponding output values (codomain) from the second row are \(y = 1, 1, 1\).

Step 3: Checking the mapping of each input

Evaluate the mapping for each input value: - For \(x = 1\), the output is \(y = 1\).- For \(x = 2\), the output is \(y = 1\).- For \(x = 3\), the output is \(y = 1\).Each value in the domain maps to exactly one value in the codomain.

Step 4: Determining if \(S\) is a function

Since each input value from the set \(\{1, 2, 3\}\) maps to one unique output, \(S\) satisfies the definition of a function.

Key Concepts

Domain and Codomain

Understanding the domain and codomain is key to grasping the nature of functions. Imagine you are organizing a party and you have a list of people you want to invite (this is the domain). Each person on this list is paired with a specific invitation detailing the time and location of the party (this is the codomain).

In mathematical terms, the domain is all possible input values. It’s like every friend you could potentially invite. In our exercise, the domain includes the numbers \(x = 1, 2, 3\). The codomain, on the other hand, is the set where all potential output values lie. It’s like a box filled with all possible invitations you might send out. Here, the codomain consists of \(y = 1\).

Every function has a specific domain and codomain, and we determine the function’s behavior by observing how the domain maps into the codomain through certain rules.

Input-Output Relationships

In a function, the input-output relationship is a fundamental aspect. It refers to the way each element of the domain (input) is paired with an element in the codomain (output). Think of it as a vending machine: every button you press (domain) is associated with a specific snack (codomain).

Let's break it down with our exercise example:

• For input \(x = 1\), the output is \(y = 1\).
• For input \(x = 2\), the output is \(y = 1\).
• For input \(x = 3\), the output is \(y = 1\).

Notice how each input has a determined, unique output. This consistent pairing is what makes the relationship a function. No matter what, each different input will provide exactly one output.

Mathematical Mappings

Mathematical mappings are crucial in defining the structural rules between inputs and outputs in a function. Each mapping represents a rule that links each element of the domain with an element of the codomain.

In our given exercise, the mapping is pretty straightforward. Every time you use an input (like pressing a vending machine button), the same output (product) is delivered consistently. The mappings are:

• \(1 \rightarrow 1\)
• \(2 \rightarrow 1\)
• \(3 \rightarrow 1\)

When a mapping like this occurs, where each input relates to exactly one output, it's clear and defined, much like matching names to hats in a coat room.

A function should have an unambiguous one-to-one relationship for every element in its domain. Each time you check an input, you will find only one output within the codomain that fits. This is what certifies the consistency and reliability of a function.