Problem 45 - Elementary and Intermediate Algebra

Question

Determine whether each equation defines \(y\) to be a function of \(x .\) If it does not, find two ordered pairs where more than one value of \(y\) corresponds to a single value of \(x .\) See Example 3. $$ x=|y| $$

Step-by-Step Solution

Answer

No, y is not a function of x because each x (e.g., x = 2) has two y values (y = 2 and y = -2).

Step 1: Understanding the Equation

The given equation is \(x=|y|\). This represents the relationship between \(x\) and \(y\) where \(x\) is the absolute value of \(y\). This means \(x\) is always non-negative.

Step 2: Analyze the Relationship

Since \(x\) corresponds to \(|y|\), for a given \(x\), there can be two possible values for \(y\): one positive and one negative. For example, if \(x = 3\), then \(y = 3\) or \(y = -3\).

Step 3: Check if y is a Function of x

In a function, each \(x\) value must correspond to exactly one \(y\) value. Here, each \(x > 0\) corresponds to two different \(y\) values (positive and negative), so \(y\) cannot be defined as a function of \(x\).

Step 4: Find Ordered Pairs with Non-Unique y

For \(x = 2\), \(y = 2\) and \(y = -2\). This results in the ordered pairs \((x, y) = (2, 2)\) and \((2, -2)\). Both these pairs demonstrate that the same \(x\) value corresponds to different \(y\) values.

Key Concepts

Understanding Absolute Value

The absolute value is a fundamental mathematical concept that refers to the distance a number is from zero on the number line. It's important to understand that the absolute value of a number is always a non-negative value, regardless of whether the original number is positive or negative.
For example:

• The absolute value of 3 is 3, denoted as \( |3| = 3 \).
• The absolute value of -3 is also 3, denoted as \( |-3| = 3 \).

This property is crucial when analyzing equations because it implies that either positive or negative numbers can yield the same absolute value. This concept was central in the given exercise where \( x = |y| \) implied that for any positive \( x \), there can be two possible values of \( y \), one positive and one negative, that lead to the same absolute value of \( x \). Understanding this helps us analyze and solve equations involving absolute values effectively.

The Role of Ordered Pairs

Ordered pairs \((x, y)\) are a fundamental concept in coordinate geometry. They provide a way to describe a relationship between two quantities, often considered as a point on a graph where the first value corresponds to the horizontal axis (often named as \( x \)), and the second to the vertical axis (\( y \)).
In the context of functions or relations, each ordered pair represents a specific correspondence between \( x \) and \( y \). This relationship can be visualized on a graph, which is particularly useful for understanding how changes in \( x \) affect \( y \).
In exercises like the one given, ordered pairs help illustrate whether a relationship is a function. For example, when \( x = 2 \), \( y \) can be 2 or -2, yielding ordered pairs \((2, 2)\) and \((2, -2)\). This shows that multiple outputs exist for a single input \( x \), which is a key insight when determining if a relation is a function.

Analyzing Relations to Define Functions

In mathematics, understanding whether a given relation is a function is vital. A function is defined as a relation where every input, or \( x \) value, corresponds to exactly one output, or \( y \) value. This principle is often called the "Vertical Line Test" because if you can draw a vertical line anywhere on the graph of the relation and it intersects the graph in more than one place, the relation is not a function.
In the given exercise, \( x = |y| \) is analyzed to determine if \( y \) is a function of \( x \). Since \( x \) can correspond to two different \( y \) values (e.g., \( y = 2 \) or \( y = -2 \) for \( x = 2 \)), this indicates that multiple outputs exist for a single input. Thus, this relation does not meet the criteria for a function.
By analyzing such relations, students can learn to correctly identify functions, which is a foundational skill in algebra and calculus. Recognizing the distinction helps in forming a proper understanding of mathematical models and their applications.