Problem 55 - Algebra 2

Question

For each relation, determine whether \(y\) is a function of \(x .\) Explain why or why not. $$ y=2 x-3 $$

Step-by-Step Solution

Answer

Yes, y is a function of x because any vertical line will intersect the graph of \(y=2x-3\) at exactly one point.

Step 1: Identify the Relation

First, we look at the equation given for the relation between variables x and y. It is given as \(y=2x-3\).

Step 2: Apply the Vertical Line Test

The Vertical Line Test helps determine if a relation is a function. If a vertical line can cross the graph of the relation at most once, then y is a function of x. Since the equation \(y=2x-3\) represents a straight line, any vertical line would intersect it at only one point.

Step 3: Determine Functionality Based on the Test

Because a vertical line would only intersect the graph of \(y=2x-3\) at one point, we can conclude that for each x-value there is exactly one y-value. Therefore, y is a function of x.

Key Concepts

Vertical Line Test

The 'Vertical Line Test' is a handy graphical technique often used to determine if a relationship between two variables constitutes a function. This test involves imagining or drawing vertical lines through the graph of the relation. If any of these vertical lines can only intersect the graph at a single point, then each input, or 'x' value, has only one output, or 'y' value, which is the defining characteristic of a function.

When applied to the equation in our exercise, \(y=2x-3\), we see that it graphs as a straight line. A straight line guarantees that no vertical line will touch the graph at more than one point. This visually confirms that we are indeed working with a function. It's an excellent quick-check method that helps students understand the one-to-one nature required for functional relationships, reinforcing the fundamental idea that a function assigns to each element in the domain exactly one element in the range.

Functionality in Relations

Understanding the 'functionality in relations' is crucial in algebra. A relation is simply a set of ordered pairs, but for it to be a function, every input (typically the x-value) must be associated with exactly one output (the y-value). This concept goes beyond just graphing and involves the core idea of deterministic relationships where one variable strictly depends on another.

In the provided exercise, the equation \(y=2x-3\) establishes a clear and predictable relationship between x and y. No matter what x-value you choose, you can always calculate the corresponding y-value using this formula, and it will be unique. This consistency is the hallmark of a function. It's important to note that not all relations are functions; some might have inputs that correspond to multiple outputs, which would fail the functionality test.

Linear Equations

Linear equations form the bedrock of algebra and are among the simplest and most widely-used mathematical tools. They describe a straight-line relationship between two variables, typically x and y, and can be written in the form \(y = mx + b\), where 'm' represents the slope and 'b' stands for the y-intercept.

The equation \(y=2x-3\) from our exercise is a linear equation with a slope of 2 and a y-intercept of -3. This slope indicates that for every one unit increase in x, y increases by two units. Linear equations are functionally significant because they graph as single, unbroken lines; this means for each x-value, there's one definitive y-value, which is exactly what defines them as functions. They serve as a great starting point for students to learn about various functional relationships and their graphical representations.