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Absolute value functions are a unique type of function where the output is always non-negative, regardless of whether the input (or the expression within the absolute value) is negative or positive. An absolute value function is typically written as \(y = |x|\), and it represents the distance from zero on the number line. This makes it an essential concept when graphing functions that involve absolute values since it directly affects the behavior or direction of the graph.

For positive numbers, \( |x| = x \).

For negative numbers, \( |x| = -x \).

In the given exercise, the absolute value function \( |x| \) is in the denominator of the expression \(f(x) = \frac{x}{|x|}\). This modifies the function into a piecewise-defined function because the behavior changes from positive to negative depending on the sign of \(x\). For of values \(x > 0\), the function evaluates to 1, as \(x\) and \(|x|\) cancel each other out. Conversely, for \(x < 0\), the result is -1. This clear switch is due to the properties of absolute value functions.

Discontinuity in a function occurs when there is a sudden jump or break in the graph. In mathematical terms, a point of discontinuity is a specific value of \(x\) where the function does not have a well-defined single value. A very clear example of discontinuity is shown in the function \(f(x) = \frac{x}{|x|}\), as detailed in the original exercise. Here, there is a break at \(x = 0\) where the function is undefined.

At \(x=0\), the absolute value makes the denominator zero, resulting in the function being undefinable at this point.

This point of undefined value means that you cannot draw the graph of the function smoothly through \(x = 0\). Discontinuities are important because they show where functions might behave unexpectedly, and it's crucial to recognize these points when analyzing or sketching graphs.

Graphing functions involves creating a visual representation of a function's behavior across different values of \(x\). It requires plotting points from a table of values, which enables a deeper understanding of how functions behave. In the original exercise, you're dealing with the function \(f(x) = \frac{x}{|x|}\).

Points were calculated and plotted for \(x\) values such as -2, -1, -0.5, 0.5, 1, and 2.

For values \(x > 0\), the points lie along the line \(y = 1\). For values \(x < 0\), the points lie along the line \(y = -1\).

The discontinuity at \(x = 0\) presents itself as a gap in the graph, illustrating that the function does not have a defined value at this point. By understanding these concepts, it becomes easier to predict the general shape and key characteristics of functions not only in simple exercises, but also in complex mathematical applications.