Problem 19 - Algebra 1: Concepts and Skills

Question

Determine whether the table represents a function. $$ \begin{array}{|c|c|} \hline \text { input } & {\text { Output }} \\ \hline 1 & {3} \\ \hline 1 & {4} \\ \hline 2 & {5} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

No, the table does not represent a function because the input 1 corresponds to two different outputs, 3 and 4.

Step 1: Check for duplicate inputs

The first step in determining if a table is a function is to look through the table and check to see if any of the input values, listed in the first column of the table, are repeated. In our given table, the input 1 is repeated.

Step 2: Check if duplicate inputs have different outputs

If there are any repeated input values, then check if these identical inputs correspond to different outputs. Looking at our table, the input value of 1 corresponds to two different output values, 3 and 4.

Step 3: Determine if the table is a function

Since in our table there is an input that corresponds to more than one output, according to the definition of a function, the table does not represent a function.

Key Concepts

Input-Output Table

An input-output table is a helpful tool to visualize relationships between sets of numbers or variables. Imagine it as a simple two-column table. The first column lists the inputs, which are the starting values or independent variables. The second column shows the outputs, which are the result of applying some operation or function to the inputs.

Understanding an input-output table is essential as it provides valuable information about the relationship between variables. You can identify patterns, trends, and even some rules that govern these relationships by studying the table. In essence, the table helps bridge the gap from the abstract notion of a function to a more concrete representation.

Definition of a Function

A function is a fundamental concept in mathematics, representing a special kind of relationship between two sets of numbers, known as the domain (inputs) and the range (outputs). A key attribute of a function is that each input is connected to exactly one output.

In simpler terms, think of a function as a machine that takes an input, performs some process, and gives an output. What distinguishes a function from other relationships is its consistency: for the same input, the function must always deliver the same output. This quality ensures predictability and stability.

To better understand this, consider the example of our table. For the table to represent a function, each input from the left column should pair with only one value in the right column. If an input leads to multiple outputs, it breaks the rule of being a function.

Duplicate Inputs

Duplicate inputs in an input-output table can be a red flag when determining whether the table represents a function. If an input appears more than once, it is crucial to check whether it is linked to the same output each time.

When duplicate inputs correspond to different outputs, as seen in our example (where input 1 maps to both 3 and 4), the table cannot be considered a function. This is because the fundamental principle of a function requires that every input is associated with only one output.

To identify a function in a table, always check the inputs first. Look for repetitions, and if they exist, examine the related outputs. Only if every duplicate input yields the same output can the table still define a function. However, if any duplicate inputs produce various outputs, as illustrated in our example, the table fails to meet the definition of a function.