Problem 69 - Algebra and Trigonometry

Question

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 2|x|+y=0 $$

Step-by-Step Solution

Answer

Yes, \( y = -2|x| \) defines \( y \) as a function of \( x \).

Step 1: Isolate the Variable y

First, we will solve the equation for \( y \). Start by subtracting \( 2|x| \) from both sides of the equation: \[ y = -2|x| \].

Step 2: Identify the Type of Function

The equation \( y = -2|x| \) describes \( y \) as being directly affected by \( x \), specifically through the absolute value of \( x \). The absolute value function \( |x| \) affects how the function behaves by ensuring that any value of \( x \) is converted into a non-negative number.

Step 3: Determine Function Criteria

For \( y \) to be a function of \( x \), each input \( x \) must produce exactly one output \( y \). The equation \( y = -2|x| \) guarantees that for every value of \( x \), there is exactly one corresponding value of \( y \). This confirms the definition of a function.

Key Concepts

Absolute Value

Absolute value is an important mathematical concept that affects how functions behave. In the equation given, we see that the absolute value notation, denoted by vertical bars such as \(|x|\), ensures that the values are non-negative. This means whether \(x\) is positive or negative, \(|x|\) turns it into its positive counterpart.

• If \(x = 3\), then \(|x| = 3\).
• If \(x = -3\), then \(|x| = 3\).

This property is crucial in solving equations because it affects how we interpret the variable increases or decreases. Absolute value functions are often used to measure distances and are foundational in other mathematical operations.

Function Criteria

Understanding whether an equation defines a function is key in many mathematical studies. For an equation to define \(y\) as a function of \(x\), each input value of \(x\) must produce only one output value of \(y\). We refer to this as the function criteria.

The original equation, after solving for \(y\), becomes \(y = -2|x|\). Here, no matter what value you plug into \(x\), the result will produce a single value for \(y\). This satisfies the criterion for being a function. In simple terms, functions always produce predictable results—a requirement for consistent mathematical modeling.

Input-Output Relationship

The concept of input-output relationships in functions explains how inputs (changes in \(x\)) affect outputs (changes in \(y\)). In the equation \(y = -2|x|\), \(x\) is the variable input that guides the changes in \(y\).

• If \(x = 1\), \(y = -2(1) = -2\).
• If \(x = -1\), \(y = -2(-1) = -2\) (still \(-2\), due to the absolute value).

This relationship helps in understanding how altering the input influences the output. In this case, multiplying the absolute value by \(-2\) ensures that \(y\) decreases as \(|x|\) increases. This consistent pattern forms the core of understanding function behaviors. By recognizing these patterns, one can interpret complex mathematical relationships more easily.