Problem 12 - Algebra 1

Question

Does the table represent a function? Explain. $$ \begin{array}{|c|c|} \hline \text { Input } & \text { Output } \\ \hline 9 & 5 \\ \hline 9 & 4 \\ \hline 8 & 3 \\ \hline 7 & 2 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

No, the table does not represent a function since the input 9 is mapped to two different outputs.

Step 1: Define Function

A function is a relation where each element in the domain (input) is related to exactly one element in the range (output). In other words, a single input cannot produce two different outputs.

Step 2: Examine the table

Look at the inputs and outputs. There is a repeated input 9 with different outputs 5 and 4. This means the input 9 is mapped to both 5 and 4.

Step 3: Determine if the table is a function

Because the input 9 has two different outputs, it violates the definition of a function that each input should map to exactly one output. Therefore, this table does not represent a function.

Key Concepts

Input and Output

When talking about functions, the concepts of "Input and Output" play a crucial role. In the context of a function, the input is the value you feed into a function, and the output is what you get after the function has processed your input. Think of it like a vending machine:

• You insert money (input).
• The machine gives you a snack (output).

In mathematics, this process is managed by rules defined by the function itself. The input is often represented by a variable such as \( x \) and the output by a variable such as \( y \). For instance, if you have the function \( y = 2x + 3 \), when you input 2, you calculate to find the output:

• Input: 2
• Calculation: \( y = 2(2) + 3 = 7 \)
• Output: 7

It is important to note that for a relationship to qualify as a function, each input should correspond to only one output.

Function Definition

A function is defined as a specific kind of relation between inputs and outputs. Every input value, known as the domain, maps to exactly one output value in the range. This unique mapping is what differentiates a function from other types of relations.
Imagine a library with only one book per shelf (output) assigned to each unique ISBN number (input). Here, no two different books (outputs) occupy the same space for a single ISBN (input).
This rule ensures that:

• Each unique input has a distinct output.
• No input is related to more than one output.

Function notation, such as \( f(x) = x + 2 \), means for every input \( x \), the function \( f \) describes exactly how to get the output, which in this case is \( x + 2 \). Solutions and calculations hinge on these clear-cut mappings to determine relationships.

Identifying Functions

Identifying whether a set of ordered pairs or a table represents a function involves checking the input-output mapping. You must ensure each input value maps to only one output value. Let's consider a completed example.
In the given table, the input values are 9, 9, 8, and 7. We need to check:

• If there is a unique output for each input.
• Whether any input repeats with different outputs.

In this case, the input "9" is connected to the outputs "5" and "4". Since one input cannot lead to different outputs in a function, this violates the function definition.
Therefore, when an input has more than one output, as in this table, it indicates the relationship is not a function. Always remember, for any input, there should be no room for ambiguity—one input, one output.