Question

Which of the tables of input-output pairs, where \(x\) represents the input and \(y\) the output, does NOT represent a function? A. \(\begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 3 & 2 & 7 & 3 & 8 & 2 \\ \hline \boldsymbol{y} & 9 & 4 & 1 & 9 & 2 & 4 \\ \hline \end{array}\) B. \(\begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 1 & 2 & 4 & 7 & 8 & 9 \\ \hline \boldsymbol{y} & 4 & 8 & 2 & 7 & 1 & 5 \\ \hline \end{array}\) C. \(\begin{array}{|c|c|c|c|c|c|c|} \hline x & 1 & 4 & 6 & 4 & 3 & 1 \\ \hline y & 2 & 3 & 8 & 7 & 6 & 2 \\ \hline \end{array}\) D. \(\begin{array}{|l|l|l|l|l|l|l|} \hline x & 5 & 8 & 2 & 4 & 2 & 5 \\ \hline y & 0 & 0 & 0 & 3 & 0 & 0 \\ \hline \end{array}\)

Short answer

Table C does NOT represent a function.

Step-by-step solution

- Understand the Definition of a Function

A function is a relation in which each input (x) has exactly one output (y). This means that for any x-value in the table, there should be only one corresponding y-value.

- Analyze Table A

Check each unique x-value to see if it maps to only one y-value. For Table A: x = 3 maps to y = 9, x = 2 maps to y = 4, x = 7 maps to y = 1, x = 8 maps to y = 2. Since each x-value has only one output, Table A represents a function.

- Analyze Table B

Check each unique x-value to see if it maps to only one y-value. For Table B: x = 1 maps to y = 4, x = 2 maps to y = 8, x = 4 maps to y = 2, x = 7 maps to y = 7, x = 8 maps to y = 1, x = 9 maps to y = 5. Since each x-value has only one output, Table B also represents a function.

- Analyze Table C

Check each unique x-value to see if it maps to only one y-value. For Table C: x = 1 maps to y = 2, x = 4 maps to y = 3 and y = 7, x = 6 maps to y = 8, x = 3 maps to y = 6. Since x = 4 maps to two different y-values (3 and 7), Table C does NOT represent a function.

- Analyze Table D

Check each unique x-value to see if it maps to only one y-value. For Table D: x = 5 maps to y = 0, x = 8 maps to y = 0, x = 2 maps to y = 0, x = 4 maps to y = 3. Since each x-value has only one output, Table D represents a function.

Key concepts

Input-Output Pairs

In algebra, functions use input-output pairs to define relationships between values.
Think of a function as a special kind of machine where each input has a unique output.
For example, if you input a number into a function, it processes this number and gives you the output value.

Each pair \(x, y\) signifies the input, denoted by \(x\), and its corresponding output, denoted by \(y\).
Consider the relationship as follows: the machine takes the input \(x\) and produces the output \(y\).

To decide if a table represents a function, ensure each input \(x\) maps to only one output \(y\).
If inputs map to multiple outputs, it contradicts the definition of a function.

Function Definition

A function is a fundamental concept in algebra that describes a relationship between two sets.
Specifically, a function assigns one unique output \(y\) to each input \(x\).

Mathematically, a function is defined as a rule or formula that takes an input from one set and gives an output in another set.

For instance, if you had a function \(f(x) = x + 2\), inputting \(3\) results in an output of \(5\) because \(f(3) = 3+2 = 5\).
Hence, this input-output pairing mechanism is what forms a function.
Key properties of functions are:

• Uniqueness of mapping: Every input maps to one unique output.
• No Repetitive Output: Each input produces a distinct output.

Unique Mapping

Unique mapping is critical in functions.
It means that each \(x\) value (input) should map to one and only one \(y\) value (output).

This uniqueness ensures a consistent and predictable relationship between inputs and outputs in a function.
For example, if you're given that \(x = 2\) maps to \(y = 4\), there is no other \(y\) value for \(x = 2\).

Let's analyze a table:
If you find for any \(x\) in the table that it maps to different \(y\) values, it indicates multiple outputs for the same input, violating the rule of unique mapping.

This property lets you ensure that a given input in a function consistently points to a single output.
Thus, verifying the uniqueness of mapping is essential to identifying a true function.