91Ó°ÊÓ

When studying functions and their graphs, identifying points of discontinuity is essential. Discontinuities occur when there is a disruption in the function's output at certain points in its domain. These can take the form of 'jumps', 'holes', or 'infinite breaks'.

In the function given by \( f(x) = \frac{|x|}{x} \), a point of discontinuity is found at \( x = 0 \). This occurs because the function is undefined at this value, resulting in a break in the graph. Specifically, evaluating the function at zero results in an indeterminate form \( \frac{0}{0} \), reinforcing the discontinuity at that point. Therefore, while the function behaves predictably for \( x > 0 \) and \( x < 0 \), \( x = 0 \) is where it "breaks" or becomes undefined.

Piecewise functions are functions that have different expressions or rules for different parts of their domain. They are used to describe situations where a function behaves differently over various intervals.

For the function \( f(x) = \frac{|x|}{x} \), it can be expressed as a piecewise function:

• \( f(x) = 1 \) if \( x > 0 \)
• \( f(x) = -1 \) if \( x < 0 \)

This means that the function outputs 1 when \( x \) is a positive number, and outputs -1 when \( x \) is negative. There is no definition given for when \( x = 0 \), which contributes to the discontinuity discussed in the previous section.

Graphical analysis involves examining the graph of a function to understand its behavior across its domain. For the function \( f(x) = \frac{|x|}{x} \), analyzing the graph can provide insights into how the function operates and where it might have discontinuities.

When graphed, \( f(x) = \frac{|x|}{x} \) appears as two distinct horizontal lines on either side of the y-axis:

• A horizontal line at \( y = 1 \) for positive \( x \)
• A horizontal line at \( y = -1 \) for negative \( x \)

There is a visible gap at \( x = 0 \) due to the function being undefined at that point. By observing the graph, the discontinuity at \( x = 0 \) becomes clear, as the lines for positive and negative \( x \) do not connect. This type of analysis is valuable in visualizing how piecewise and discontinuous functions behave.