Problem 11 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the relation represents \(y\) as a function of \(x .\) Explain your reasoning. $$\begin{array}{|l|l|l|l|l|l|} \hline \text { Input, } x & -3 & -1 & 0 & 1 & 3 \\ \hline \text { Output, } y & -9 & -1 & 0 & 1 & -9 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

Yes, \( y \) is a function of \( x \) because every input \( x \) has exactly one output \( y \).

Step 1: Observe the table

Examine the given table, specifically the output \( y \) for each corresponding input \( x \). In the given table, we have \( x \) values of -3, -1, 0, 1, and 3 and corresponding \( y \) values of -9, -1, 0, 1, and -9 respectively.

Step 2: Check for repeated inputs

Look for any repeating \( x \) values and observe their corresponding \( y \) output. Repeated \( x \) values are not a problem as long as they map to the same \( y \) value every time.

Step 3: Confirm if \( y \) is a function of \( x \)

Having ascertained that there is no repeating \( x \) value in the table, it is now evident that no \( x \) value has more than one corresponding \( y \) value. Therefore, it can be concluded that \( y \) is definitely a function of \( x \).

Key Concepts

Relation Represents Function

In mathematics, understanding how a relation represents a function is crucial. A function is essentially a special type of relation where each input is associated with exactly one output.
For example, if we consider pairs of numbers where the first number is an 'input' and the second one is an 'output', if each input has only one corresponding output, we have a function. This fits the definition of a function as a relation that assigns a single value of the dependent variable, often denoted as 'y', for every value of the independent variable, frequently represented as 'x'.
Looking at the provided exercise, we see a list of inputs (x) and their corresponding outputs (y). If we were to draw these as points on a graph, a function would mean that every vertical line drawn through a graph of the relation would intersect the relation at a maximum of one point. This description is often referred to as the 'vertical line test'. The given table passes this test, hence the relation represents a function.

Determining Functions

Determining whether a relation is a function involves checking the pairing of inputs to outputs to ensure uniqueness. This process is a fundamental concept in algebra.
As indicated in the solution's Step 2, a key step is to inspect for repeated inputs; if discovered, they must map to the same output value. This ensures each input has only one outcome—fulfilling the basic requirement of a function. If an input corresponds to more than one output, the relation is no longer a function.

• Observe all input-output pairs
• Check for unique outputs for each input
• Apply the vertical line test*

By following these steps, one can effectively determine if a relation qualifies as a function. In the exercise, each input has only one corresponding output, which confirms that we indeed have a function. Note that the 'vertical line test' is a visual method applicable to graphical representations of relations.

Input-Output Relationship in Functions

The input-output relationship is the backbone of the function concept in mathematics. The 'input' represents the independent variable, while the 'output' is the dependent variable. For each input, there is a rule that determines exactly one output.
The relationship is often envisioned as a machine where every time you insert an input (drop in a coin), you receive an output (get a gumball). By definition, a function ensures that no matter how many times you input the same 'coin', the 'machine' will always provide the same 'gumball'. There's no randomness or uncertainty—this predictability is what makes a rule a function.

Considering the Exercise

With the provided table of values, we confirm the input-output relationship adheres to this predictable pattern, solidifying the rule as a function.

Function Definition in Algebra

In algebra, a function is defined as a rule that relates each element in a set, known as the domain (typically the set of all possible inputs), to exactly one element in another set, known as the range (the set of all possible outputs).
Functions can be represented in various ways including equations, graphs, and tables. The notation usually involves terms like 'f(x)' which denotes a function named 'f' with 'x' as the input variable. This representation conveys that for every value of 'x', 'f(x)' will yield a corresponding value.

Relationship with the Exercise

The table in the exercise exemplifies one way to visualize a function—mapping discrete 'x' values to 'y' values. Here, each value of 'x' corresponds to exactly one value of 'y', aligning perfectly with the function definition in algebra.