Problem 90 - College Algebra with Modeling and Visualization

Question

Determine if \(y\) is a function of \(x\). $$ y=|x| $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Question

We need to determine if the expression given, \( y = |x| \), defines \( y \) as a function of \( x \). This means that for each input \( x \), there should be exactly one output \( y \).

Step 2: Analyzing Absolute Value

In the expression \( y = |x| \), \(|x|\) represents the absolute value of \( x \). The absolute value operation reflects all negative numbers into positive, such that \(|x| = x\) if \(x\geq 0\) and \(|x| = -x\) if \(x < 0\).

Step 3: Checking for Multiple Outputs

A key property of a function is that each input \( x \) produces exactly one output \( y \). For \( y = |x| \), for any given number \( x \), there is only one corresponding value of \( y \) because the absolute value function always outputs a single, non-negative number regardless of the sign of \( x \).

Step 4: Conclusion

Since \( y = |x| \) gives exactly one \( y \) value for each \( x \), \( y \) is indeed a function of \( x \).

Key Concepts

Absolute Value

The concept of absolute value is an important foundational mathematical idea. The absolute value of a number is essentially the "distance" of that number from zero on the number line, without considering direction. This is why, regardless of whether a number is positive or negative, its absolute value is always expressed as a non-negative number.

In more formal terms, the absolute value of a number \(x\) is represented by \(|x|\). Here's a quick breakdown:

• If \(x\) is positive, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).
• If \(x\) is zero, \(|x| = 0\).

Understanding this operation is key to grasping why functions like \(y = |x|\) behave the way they do. By transforming negative inputs into positive outputs, absolute value ensures that each input has a single corresponding output, simplifying the input-output relationship of the function.

Input-Output Relationship

A fundamental property of any function is its input-output relationship. This relationship involves a set of possible inputs (often denoted as \(x\)) and outputs (denoted as \(y\)). For a set of ordered pairs to qualify as a function, each input must correspond to exactly one output.

Let's consider the function described by \(y = |x|\). Here, no matter what value \(x\) takes—negative, positive, or zero—there is always one, and only one, corresponding value for \(y\). This is why all points on the graph of \(y = |x|\) are unique for each \(x\).

This unique pairing in the function ensures that if you know the input \(x\), you can always determine the exact output \(y\). This characteristic is essential for defining \(y\) as a function of \(x\), as it assures consistency and predictability in its behavior.

Function Definition

In mathematics, a function is a special kind of relation between sets—the set of inputs and the set of possible outputs. For a relationship to be classified as a function, every input must have exactly one output. This is known as the "vertical line test" in graphical terms—if no vertical line intersects the graph of a relation at more than one point, it's a function.

The equation \(y = |x|\) meets the criteria for being a function. Why? Because for every value of \(x\), including both positive and negative numbers and zero, \(y\) has a unique absolute value. This means every input produces one and only one output.

• For instance, if \(x = 2\), then \(y = 2\).
• If \(x = -3\), \(y\) still equals 3.

This precise input-output consistency is what makes \(y = |x|\) a function of \(x\). Functions like these are integral to understanding many core mathematical concepts and can help to predict and model real-world phenomena accurately.