Problem 36 - 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 \(x .\) $$ y=|x| $$

Step-by-Step Solution

Answer

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

Step 1: Understand the definition of a function

A function is a relation in which each input value (commonly denoted as \(x\)) is associated with exactly one output value (commonly denoted as \(y\)). We need to determine if for every \(x\), there is only one corresponding \(y\).

Step 2: Analyze the given equation

The equation given is \(y = |x|\). The absolute value function takes any real number \(x\) and gives its non-negative value as output. This means for each \(x\) value, there is a unique and single \(y\) value which is always non-negative.

Step 3: Determine if \(y = |x|\) is a function

Since the absolute value function produces one and only one outcome for every input \(x\), there cannot be more than one \(y\)-value for any \(x\)-value. Therefore, \(y = |x|\) satisfies the definition of a function.

Key Concepts

Absolute Value Function

The absolute value function is a special kind of mathematical relationship that transforms any real number into its non-negative counterpart. Imagine your friend owes you money. Whether your friend owes you \(5 or \)-5, all you care about is the absolute amount owed, which is $5. That's like the absolute value. It takes any number and makes it positive.

Mathematically, we denote the absolute value of a number as \(|x|\). This function is defined for any real number as follows:

• If \( x \geq 0 \), then \( |x| = x \)
• If \( x < 0 \), then \( |x| = -x \)

This simple rule ensures that all outputs of the absolute value function are zero or positive, making it a neat tool to express distance or magnitude, regardless of direction. So, when the equation \( y = |x| \) is considered, for every value of \( x \), there is a single output \( y \), maintaining the function's integrity.

Relation

Relations are a fundamental concept in algebra that describe how two sets of numbers or objects are connected. Think of a relation as a social media network, where users (inputs) have connections (outputs). Each input from one set is paired with outputs from another set. However, these pairings can be more flexible than functions.

In mathematical terms, a relation can map one input to multiple outputs or even no outputs at all. For example, let's take a simple mathematical operation like squaring a value. If we have the relation \( y = x^2 \), you can see that both \( x = 3 \) and \( x = -3 \) result in \( y = 9 \).

While all functions are relations, not all relations are functions. Functions are special sorts of relations where each input is connected to exactly one output. When assessing whether a relation is a function, ask yourself: "For each x, do I get only one y?" That is the key distinguishing factor.

Input-Output Relationship

Understanding input-output relationships is essential in math and real life. These relationships tell us how a certain input transforms into an output. Consider a coffee machine. You put in a pod (input), press a button, and get coffee (output). This describes a clear input-output relationship.

In mathematics, input-output relationships are expressed through functions. In the function \( y = |x| \), \( x \) is the input variable, and \( y \) is the output variable. The function tells us exactly how to transform any given \( x \) into \( y \). For every particular number you input, there's one output number associated with it.

Being able to predict outputs based on inputs is a powerful tool in problem-solving and analysis. It allows us to understand and forecast behaviors in systems ranging from simple mechanical devices to complex scientific models. In functions, this predictability comes from the fact that each input corresponds to exactly one output, ensuring consistency and reliability in expectations.