91Ó°ÊÓ

The absolute value of a number is its distance from zero on the number line. In mathematical terms, the absolute value is always a non-negative number. For any real number \( x \), the absolute value is represented as \( |x| \). This notation means that:

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

This property is what makes the absolute value function unique. It ensures that no matter whether \( x \) is positive or negative, the output (\( y \) in the equation \( y = |x| \)) remains positive or zero. This characteristic simplifies many algebraic expressions and makes the absolute value a fundamental concept in mathematics. As the function \( y = |x| \) assigns exactly one \( y \)-value for each \( x \)-value, it congruently meets the definition of a mathematical function.

The vertical line test is a straightforward visual method for verifying whether a curve on the graph represents a function. The test states that a curve is a graph of a function if no vertical line intersects the curve more than once at any given \( x \)-coordinate.When you apply this test to the equation \( y = |x| \), you'll see that for every vertical line drawn over any \( x \)-value, the line crosses the graph only once. This means every possible \( x \) corresponds to a single \( y \). It verifies that \( y = |x| \) is a function.
This technique is particularly useful because it provides a quick check: if any vertical line crosses your curve multiple times, the graph does not describe a function. But for \( y = |x| \), the V-shaped graph will always meet our evaluations, showcasing its function status succinctly.

A \( function \) is a special kind of relation where each input value has exactly one output value.

• The key rule for functions is that for each \( x \)-value (input), there must be exactly one \( y \)-value (output).
• This assures no ambiguity: every time you input a certain \( x \), you know exactly what \( y \) will be.

In the context of the given problem, the equation \( y = |x| \) perfectly satisfies the criteria of a function because:

• It consistently provides a single output \( y \) for each possible \( x \).
• Whether \( x \) is positive, negative, or zero, \( |x| \) computes to one definite, predictable \( y \).

This fittingly aligns with the function definition, ensuring clarity and precision in mathematical operations involving \( y = |x| \). Functions are foundational in algebra and calculus, helping to map relationships and predict outcomes accurately.