Problem 65 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 9.) \(2|x|+y=0\)

Step-by-Step Solution

Answer

Yes, the equation defines y as a function of x.

Step 1: Isolate y in the Equation

Start by isolating \( y \) in the equation. The original equation is \( 2|x| + y = 0 \). To isolate \( y \), subtract \( 2|x| \) from both sides: \( y = -2|x| \).

Step 2: Determine if a Single Output Exists for Each Input

Now that \( y \) is isolated, check if for each value of \( x \) there is only one possible value for \( y \). Since \( y = -2|x| \) simply involves calculating the absolute value of \( x \), multiplying by \(-2\), and only results in one value for \( y \), for each \( x \) there is only one corresponding \( y \).

Step 3: Conclusion about Function Definition

Since each \( x \) value results in a unique \( y \) value, the equation defines \( y \) as a function of \( x \). The equation meets the conditions of a function because each input \( x \) has exactly one output \( y \).

Key Concepts

Isolating Variables

When working with equations in algebra, one common step is isolating a variable, usually to solve for that variable or determine its role in the equation. Isolating a variable means rearranging the equation so that the variable you are interested in stands alone on one side of the equation. For instance, in the equation \( 2|x| + y = 0 \), the goal was to isolate \( y \). This involved subtracting \( 2|x| \) from both sides which resulted in \( y = -2|x| \). By isolating \( y \), we can better understand its relationship with \( x \). This step converts complex expressions into simpler, more manageable forms, enabling further analysis, such as checking if \( y \) is a function of \( x \).

• Identify the variable to isolate.
• Use arithmetic operations to rearrange the equation.
• Ensure the isolated form maintains equality.

Isolation of variables is a foundational skill that facilitates understanding and solving mathematical problems efficiently.

Absolute Value

The absolute value is an important concept in math that refers to the non-negative value of a number without regard to its sign. It is represented by two vertical bars surrounding the number or expression, such as \(|x|\). For example, \(|-3| = 3\) and \(|3| = 3\). In the context of the equation \( y = -2|x| \), the absolute value affects how \( x \) influences \( y \), since it ensures that the expression inside is always non-negative.Understanding absolute value is essential because:

• It represents the distance of the number from zero on a number line.
• It simplifies the analysis of equations by ignoring the sign.
• It allows equations with potentially two solutions to provide a single, non-negative outcome.

This ensures that no matter whether \( x \) is positive or negative, \( y \) only sees the magnitude of \( x \), contributing to consistent and predictable results.

Unique Outputs

When discussing functions, a key characteristic is whether each input corresponds to exactly one output. This property defines a relationship as a function. In mathematical terms, if for each \( x \) there is one distinct \( y \), the equation describes \( y \) as a function of \( x \). In the equation \( y = -2|x| \), each value of \( x \) results in one unique corresponding value for \( y \). This occurs because, despite the absolute value transformation, each input affects the output predictably: the equation remains consistent and deterministic. Certain features of unique outputs include:

• Each input value maps to exactly one output.
• Eliminates ambiguities in the equation's outputs.
• Validates the conditions that define a function.

This unique input-output correspondence confirms the function definition, making it a crucial aspect of understanding relationships in algebra.