Problem 11 - Algebra 1

Question

Does the table represent a function? Explain. $$ \begin{array}{|c|c|} \hline \text { Input } & \text { Output } \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline 3 & 11 \\ \hline 4 & 18 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Yes, the table represents a function. This is because each input has only one corresponding output.

Step 1: Identifying the inputs and outputs

Firstly, identify all inputs (in the left column) and their corresponding outputs (in the right column) from the table. The table presents the following pairs: (1, 3), (2, 6), (3, 11), and (4, 18).

Step 2: Checking if each input has only one output

Now examine each input and check if they correspond to only one output. Here, each input 1, 2, 3, 4 corresponds to exactly one output 3, 6, 11, 18 respectively.

Key Concepts

Input-Output Relationships

In mathematics, when we talk about functions, one of the most important concepts is the input-output relationship. This is what makes a function capable of transforming data. The input is like the "ingredient" and the output is the "result" we get after processing the ingredient using a function.

Functions are like machines in which we feed some values (inputs) and get some results (outputs). Each specific input should provide us with a specific output, showcasing a clear one-to-one relationship.

• Each input corresponds to one output.
• If an input has two different outputs, it is not a function.
• The table of values helps visualize these relationships.

This simple, yet powerful concept allows us to understand complex systems by breaking them down into easily manageable parts.

Function Definition

Defining a function involves understanding the rule that links inputs to outputs. A function assigns each element from a set of inputs to exactly one element of a set of outputs.

In simpler terms, for each item or number we put in, we get out a specific outcome, like a predictable pattern. This means if we input the value of 2 and receive an output of 6, the function definition makes this consistent every time we input 2.

• A function takes inputs from a specific domain and maps them to outputs in a range.
• It must provide a single output for each input to be classified as a function.
• This consistency is what differentiates a function from other mathematical relations.

By following these rules, we can determine whether a table or list of ordered pairs is representing a function by checking its inputs and ensuring they lead to only one output each time.

Tables of Values

When exploring functions, tables of values are incredibly useful tools as they allow us to present and examine input-output relationships systematically. These tables set out pairs like (input, output), clearly showing how each input has its corresponding output.

For example, in our previous task, the table provided us with pairs like (1, 3) and (2, 6). As we look down the inputs and corresponding outputs, we confirm that each input number leads only to one output.

• Tables help organize data into clear inputs and outputs.
• They provide a visual representation of whether a relationship is a function.
• By checking the table, we can quickly ascertain if each input has a unique output, which is the hallmark of a function.

Using tables of values makes it easier to validate the function's definition and ensure that it adheres to the necessary mathematical properties.