Problem 81 - College Algebra with Modeling and Visualization

Question

Determine if \(S\) is a function. \(S\) is given by the table. $$ \begin{array}{cccc} x & 1 & 3 & 1 \\ y & 10.5 & 2 & -0.5 \end{array} $$

Step-by-Step Solution

Answer

No, \(S\) is not a function as \(x = 1\) maps to two different \(y\) values.

Step 1: Understand the Definition of a Function

A function is defined as a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. This means that for every value of \(x\), there should only be one corresponding \(y\) value.

Step 2: Analyze the Table Data

From the table, observe the \(x\) and \(y\) values:- \(x = 1\) maps to \(y = 10.5\).- \(x = 3\) maps to \(y = 2\).- \(x = 1\) maps to \(y = -0.5\).

Step 3: Check for Repeated \(x\) Values with Different \(y\) Outputs

Notice that \(x = 1\) appears twice in the table but maps to two different \(y\) values: \(y = 10.5\) and \(y = -0.5\).

Step 4: Determine if \(S\) is a Function

Since there is an \(x\) value (in this case, \(x = 1\)) that does not consistently map to a single \(y\) value, \(S\) is not a function according to the definition.

Key Concepts

Input-Output Relation

The concept of input-output relation in mathematics is a fundamental element that helps us understand the behavior of functions. In simple terms, think of it as a matching game where every input must correspond to one single output. Picture a machine where you put something in, and one predictable outcome emerges. Input-output relationships can be seen in everyday situations, like pressing a button to turn on a lamp, where pressing is the input, and the light turning on is the output.

When we analyze these relationships in mathematics, we typically use variables like \(x\) for inputs and \(y\) for outputs. For a relation to qualify as a function, each \(x\) (input) must pair with only one \(y\) (output). If the same \(x\) produces more than one \(y\), it breaks the rule of being a function. Hence, ensuring a single output for each input is crucial for understanding whether we are dealing with a true function.

Function Definition

Understanding the definition of a function is essential for recognizing one in a problem. A function is a type of relation where every input of the domain (which is all possible \(x\) values) is related to exactly one output in the codomain (all possible \(y\) values). This means that if any \(x\) is mapped to two different outputs, the relation can no longer be termed as a function.Here’s a simple way to grasp it:

• Think of a function as a set of rules or a system, like a vending machine, that gives you a consistent product (output) for a specific code entered (input).
• If you enter the same code twice expecting the same product, but get different ones, the machine isn’t functioning properly — it’s similar to a failing function in math.
• In our example, since \(x = 1\) relates to both \(y = 10.5\) and \(y = -0.5\), this violates the rule of a function.

So, when determining if something is a function, checking that every input is linked to one output is the key step.

Table Analysis

Analyzing tables is a strategic approach to drive clarity into understanding the input-output connections within a potential function. Each row or column (depending on setup) typically lists a set of \(x\) values (inputs) and corresponding \(y\) values (outputs). This clear representation helps quickly identify whether each input has a single, unique output.Let’s break it down:

• Look for repeated \(x\) values with differing \(y\) outputs. If found, like in our example where \(x = 1\) is paired with both \(y = 10.5\) and \(y = -0.5\), this signals that it is not a function.
• Each unique \(x\) must map consistently to the same \(y\) to maintain function status.
• Tables simplify the checking process by laying out all input-output pairs clearly, making it easier to spot errors in function adherence.

By carefully examining a table, you can quickly decide if the criteria of a function are met, ensuring every input has just one output that adheres to the rules of what defines a mathematical function.