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Understanding the concept of *absolute value* is crucial when dealing with functions that include it. The absolute value of a number is its distance from zero on a number line. This means it is always a non-negative number. For example, the absolute value of both 3 and -3 is 3.
In mathematical notation, the absolute value of a number, say \(a\), is written as \(|a|\). This indicates that \(|-x| = x\), because distance cannot be negative. In the context of our exercise, where \(y = |4 - x|\), the absolute value ensures that any evaluation of \(4 - x\) will result in a non-negative number.
The absolute value function is often visualized as a V-shape in graph plots, reflecting how the values "bounce" off the x-axis, just as negative values turn positive. This feature makes it unique among linear functions.

The backbone of many mathematical analyses, *linear equations* are those that graph as straight lines. A typical form of a linear equation in two variables, \(x\) and \(y\), is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
When analyzing \(y = |4 - x|\), the expression inside the absolute value braces, which is \(4 - x\), itself represents a linear equation: it is of the form \(y = -x + 4\). This suggests that this is a line with a slope of -1 and a y-intercept of 4. However, when wrapped in the absolute value, it becomes a V-shaped graph, indicative of an absolute value function.
This transformation occurs because each linear piece on either side of 4 is reflected to maintain only non-negative y-values, which is a hallmark of absolute value functions. Understanding this change is vital to grasping how absolute value modifies linear equations.

*Function testing* involves checking whether a particular equation defines a function. By definition, a function implies that each input corresponds to precisely one output. So, if any input \(x\) results in multiple values of \(y\), the equation is not a function.
For the equation \(y = |4 - x|\), the input \(x\) yields only one output for every \(x\) value. This behavior conforms to the definition of a function. Essentially, the inclusion of the absolute value in the equation guarantees that no matter what \(x\) is, the resulting \(y\)-value will be unique due to the nature of absolute value not allowing negative results.

• If \(x = 3\), then \(y = |4 - 3| = 1\).
• If \(x = 5\), then \(y = |4 - 5| = 1\).

These examples show consistent one-to-one correspondence between input and output, confirming the function status of \(y = |4 - x|\). Thus, through function testing, we affirm that \(y\) is indeed a function of \(x\).