Problem 16 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

Yes, the table represents a function because each input is matched with exactly one output.

Step 1: Identify the Inputs and Outputs

The inputs in this problem are the numbers in the left column of the table (1, 2, 3) and the outputs are the numbers in the right column (3, 4, 5).

Step 2: Check for Repeated Inputs

Look at the list of inputs and determine whether any numbers are repeated. In this case, all the input numbers 1, 2, and 3 appear only once. There are no repeated inputs.

Step 3: Determine Whether the Table Represents a Function

Since there are no repeated inputs in the table, it follows that each input is matched with exactly one output. Therefore, the table does represent a function.

Key Concepts

Inputs and Outputs

In mathematics, especially in functions, **inputs** are the values that you plug into a function, while **outputs** are the results you obtain after applying the function rules to the inputs. Think of a function as a machine:
- You insert an input. - The machine follows specific rules. - Out comes the output.

For instance, imagine a function that doubles any number you give it. You give it an input, say 4, and it produces an output of 8.
- **Input**: 4 - **Output**: 8

However, it's important to understand that for something to be a function, each input must have exactly one output. Therefore, if you input a 4 into our machine and sometimes get 8 and other times 10, we do not have a proper function.

Function Table

A function table displays the relationship between inputs and outputs. It typically consists of two columns: the first for inputs and the second for outputs. When examining a function table, the goal is to determine whether each input is paired with exactly one output.
For example, consider this table:

• Input: 1, Output: 3
• Input: 2, Output: 4
• Input: 3, Output: 5

To decide if such a table represents a function, check to see if any input appears more than once. If no input is repeated, you have a function. In our example, inputs (1, 2, 3) each occur only once. Thus, it confirms the table is a function.
Keep in mind that if any input is paired with more than one output, then it cannot be a function.

Algebra

Algebra often involves using functions to express relationships between variables. It is a branch of mathematics where numbers are represented by letters to generalize and solve problems more easily.
When dealing with functions in algebra, you frequently encounter function expressions, like \( f(x) = 2x + 3 \). This expression tells you:

• The function's name is "f".
• For any input \( x \), the function returns an output after executing the operation.

Each \( x \) in this context represents an input, and the output is calculated by multiplying the input by 2 and then adding 3. Suppose you choose an input value of 2. Plugging it into the expression \( f(2) = 2 \times 2 + 3 \) results in an output of 7.
This algebraic approach to functions is crucial as it allows you to model real-life scenarios and solve complex problems efficiently.