Question

Determine whether the equation defines y as a function of \(x .\) \(y=|x|\)

Short answer

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

Step-by-step solution

Identify the given equation

The given equation is \(y = |x|\). This equation involves the absolute value of \(x\).

Understand the definition of a function

A function is a relation in which each input (\(x\)) has a unique output (\(y\)). In other words, for every value of \(x\), there should be exactly one corresponding \(y\).

Analyze the absolute value function

The absolute value of \(x\), denoted \(|x|\), is always a non-negative value. It means that \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).

Evaluate the uniqueness of outputs for each input

For any given \(x\), \(|x|\) will always produce a single value. For example, if \(x = 3\), \(|3| = 3\). If \(x = -3\), \(|-3| = 3\).

Conclusion

Since the absolute value function \(y = |x|\) produces a unique value for every input \(x\), the equation defines \(y\) as a function of \(x\).

Key concepts

Relation and Function

Let's start by understanding what a function is. In mathematics, a function is a way of matching elements from one set (called the domain) to elements in another set (called the range). A relation is any pairing of elements from one set to another. Functions are special types of relations where each input has exactly one output. This means if you have an equation like \(y = |x|\), each value you plug in for \(x|\) gives you only one specific value for \(y|\). This unique pairing is what makes an equation a function. It ensures that there are no ambiguities or multiple outcomes for any given input.

Unique Output

Now let's go deeper into the idea of a unique output. When we say that a function provides a unique output for each input, we're essentially stating the distinctiveness rule of functions. Consider the equation \(y = |x|\). If you input \(x = 2\), the absolute value is \(2\), and \(y\) will also be \(2\). If you input \(x = -2\), the absolute value is still \(2\), hence \(y\) will again be \(2|\). No matter what value you choose for \(x|\), \(y|\) always ends up being a single, specific number, thereby providing a unique output. This confirms that our equation \(y = |x|\) behaves as a function since it assigns one and only one value to \(y|\) for every \(x|\).

Non-Negative Value

One interesting feature of the absolute value function is that it always produces non-negative values. The absolute value of a number is its distance from zero on the number line, regardless of direction. For the function \(y = |x|\), whether \(x\) is positive, negative, or zero, \(y\) will always be either positive or zero. This is because the absolute value converts negative inputs to positive outputs. If \(x\) is positive or zero, \(y\) simply equals \(x\). If \(x\) is negative, \(y\) equals the positive counterpart of \(x\). For example: \(y = |-3| \) becomes \(y = 3\). No input will ever yield a negative value for \(y|\). This non-negative property is a key component in identifying the nature of absolute value functions.