Problem 18 - Algebra 1: Concepts and Skills

Question

Determine whether the table represents a function. $$ \begin{array}{|c|c|} \hline \text { input } & {\text { Output }} \\ \hline 2 & {2} \\ \hline 3 & {4} \\ \hline 4 & {6} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Yes, the given table does represent a function, as each input has a single, unique output.

Step 1: Identify Inputs and Outputs

Review the table and identify all of the inputs and the corresponding outputs. Here, the inputs listed in the table are 2, 3, and 4. And their corresponding outputs are 2, 4, and 6 respectively.

Step 2: Checking for Unique Outputs

Check if each input has only one unique output. In the given table, the input 2 has an output of 2, input 3 has an output of 4 and input 4 has an output of 6. Clearly, for each input there is only one unique output.

Step 3: Confirm the Function

Since each input has exactly one unique output, the table does represent a function according to the definition of function in mathematics.

Key Concepts

Functions in Mathematics

When we talk about functions in mathematics, we're looking at a specific type of relationship between two sets, typically referred to as the domain (inputs) and the range (outputs). A function assigns to each element in the domain exactly one element in the range. This is like saying for every x, there is only one y that corresponds to it.

Think of a function as a special machine where you put in a number, and a specific operation is performed to give another number. Now, this function-machine is consistent; if you input the same number, you always get the same result. The concept of a function is fundamental in various areas like calculus, algebra, and even advanced fields like complex analysis and differential equations.

Input-Output Relationship

Understanding the input-output relationship is crucial in evaluating whether a set of pairs correctly represents a function. In simpler terms, for each input, there must be a well-defined output. Returning to our function-machine analogy, when you input a particular value, the machine should produce one and only one output.

To examine if the given pairs adhere to this criterion, we identify the set of inputs and their corresponding outputs. If we find that an input corresponds to more than one output, the relationship breaks the rule of functions, and therefore, it isn't a function. In the exercise provided, we see that there's a clear rule linking inputs to outputs, with no input relating to multiple outputs.

Unique Outputs

The hallmark of a function is its unique outputs: every single input yields one specific output, never more. This is non-negotiable in the realm of functions. It's this exclusivity that allows functions to have graphs that pass the 'vertical line test.' When you draw a vertical line through the graph of a relationship, if it touches the graph at more than one point, then the relationship is not a function.

In the context of our exercise, we apply the idea of unique outputs by checking each input against its output. The table given fulfills this essential criterion as every input is matched with one unique output – no repetitions or duplications occur. This is a green light signalling that indeed, the table represents a function.