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Absolute value functions are unique due to their property of always producing non-negative results. The absolute value of a number, denoted as \(|x|\), is defined as the distance from zero on the number line, regardless of direction. This means that for any real number \(x\), \(|x|\) is always non-negative.

When dealing with the function \(G(t) = 1/|t|\), the absolute value affects how the function is defined. Since the absolute value \(|t|\) is in the denominator, it's crucial to recognize that \(|t|\) cannot be zero, or the function would be undefined. Therefore, \(t\) cannot be zero.

Both positive and negative values of \(t\) have to be considered separately in such functions. When \(t > 0\), \(|t| = t\), and the function simplifies to \(1/t\). Conversely, when \(t < 0\), \(|t| = -t\), making the function \(-1/t\). These distinctions help determine the behavior of the function across its domain.

The concept of asymptotes is important in understanding functions like \(G(t) = 1/|t|\). Asymptotes are lines that a graph approaches but never actually meets. In this case, \(G(t)\) has a vertical asymptote at \(t = 0\) because the function is undefined at that point. This means the graph will never touch or cross this vertical line.

The behavior of the function near the asymptote provides insight into its graphing and helps predict how the function behaves as the variable grows larger or smaller in the positive or negative direction.

For \(t > 0\), as \(t\) decreases towards zero, the function value increases without bound, indicating the presence of a vertical asymptote. Similarly, for \(t < 0\), as \(|t|\) decreases towards zero, the function \(-1/t\) grows negatively towards infinity, following the same asymptotic behavior.

Graphing functions like \(G(t) = 1/|t|\) combines understanding absolute values and asymptotes. This particular graph consists of two distinct branches: the first quadrant where \(t > 0\), and the second quadrant where \(t < 0\). Because the function is undefined at \(t = 0\), these branches never meet, creating a gap in the graph.

As \(t\) becomes larger in the positive direction, the branch in the first quadrant approaches the horizontal axis but never touches it. This results in a curve that heads towards zero as \(t\) increases. Similarly, for \(t < 0\), as \(|t|\) increases, the branch in the second quadrant also approaches the axis, creating a symmetric behavior reflected about the t-axis.

This symmetry and the presence of the asymptote at \(t = 0\) are key to understanding how the graph behaves and helps in sketching it accurately. The shape of the graph indicates the distinct behaviors of the function over different intervals, ensuring a full understanding of the function's essence through visualization.