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To understand a function fully, creating its graph can be incredibly helpful. A graph visually represents the behavior of a function over a range of values.
In the case of the function \(g(x) = \frac{|x|}{x^{2}}\), the graph consists of two separate parts due to the absolute value and the quadratic components. There is a critical point at \(x = 0\) where the function is undefined. This leads to a vertical asymptote, which the graph approaches but never touches or crosses.

• For \(x > 0\), the graph depicts the function \(y = \frac{1}{x}\), which is a hyperbola situated in the first quadrant.
• For \(x < 0\), the graph changes to \(y = -\frac{1}{x}\), creating another hyperbolic shape in the third quadrant.

Understanding these segments helps in sketching an accurate graph of the function, making it clear that the value \(x = 0\) is not within the domain due to the discontinuity.

Absolute value functions introduce unique attributes to functions due to their nature of always being non-negative. The absolute value of \(x\), written as \(|x|\), equals \(x\) if \(x\) is positive or zero, and \(-x\) if \(x\) is negative.
In the expression \(g(x) = \frac{|x|}{x^{2}}\), the absolute value affects the behavior depending on whether \(x\) is positive or negative, ensuring that the numerator is always positive for positive \(x\) or negative \(x\).
This results in:

• \(g(x) = \frac{1}{x}\) when \(x > 0\).
• \(g(x) = -\frac{1}{x}\) when \(x < 0\).

This symmetry around the y-axis influences the graph and behavior of the function. In graphing, the absolute value assures symmetry across specific axes, often leading to distinct characteristics such as peaks or reflections.

Understanding the behavior of a function involves exploring how it behaves over its domain. For \(g(x) = \frac{|x|}{x^{2}}\), the behavior changes depending on whether \(x\) is positive or negative.
Key observations help depict its behavior efficiently:

• For \(x > 0\), \(g(x)\) simplifies to \(\frac{1}{x}\), showing a decreasing curve from the positive y-axis towards zero but never reaching it, representing an asymptotic behavior.
• For \(x < 0\), \(g(x)\) becomes \(-\frac{1}{x}\), indicating an increase towards zero from negative y-values, but similarly never reaching zero.
• There's an important exclusion at \(x = 0\), since the function is undefined due to division by zero. This point forms a significant break, called a discontinuity. As \(x\) approaches zero from either direction, the values of \(g(x)\) become exceedingly large in magnitude, indicating the presence of a vertical asymptote.

This comprehensive understanding of how \(g(x)\) behaves aids in creating an accurate graph and interpreting the related dynamics of the function.