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A piecewise function is a type of function characterized by different expressions based on particular intervals of the input value, usually denoted by the variable \(x\). In the context of the given function \(f(x) = x \cdot |x|\), it can be broken down into distinct functions that are applicable when \(x\) is greater than or equal to zero and when \(x\) is less than zero.

• For \(x \geq 0\), the absolute value \(|x|\) equals \(x\), thus the function simplifies to \(f(x) = x^2\).
• For \(x < 0\), the absolute value \(|x|\) equals \(-x\), resulting in the function \(f(x) = -x^2\).

This segmentation allows us to analyze and define the behavior of the function at different intervals clearly. When graphing or analyzing such functions, knowing how the function splits is crucial to understanding the overall behavior of the function and how each piece behaves independently.

Graphing functions is a vital skill in mathematics that involves translating the function's equation into a visual representation on the coordinate plane. For our specific piecewise function, \(f(x) = x \cdot |x|\), we deal with two separate parts of the function:

• The graph of \(f(x) = x^2\) for \(x \geq 0\) is a parabola that opens upward, originating from the point \( (0,0) \).
• Conversely, the graph of \(f(x) = -x^2\) for \(x < 0\) is a downward-opening parabola, also meeting at the origin \( (0,0) \).

When plotting these on a coordinate grid, it's essential to carefully consider each interval to create an accurate representation of the function. Both parabolas are smoothly connected at the origin, depicting how the function behaves towards the positive or negative x-axis. This makes visual analysis like the horizontal line test possible, which is essential for identifying certain properties related to the function.

The horizontal line test is a simple method used to determine if a function is one-to-one. A function is considered one-to-one if and only if no horizontal line intersects the graph more than once.

For the function \(f(x) = x \cdot |x|\), the graph consists of an upward parabola for \(x \geq 0\) and a downward parabola for \(x < 0\). These sections intersect the y-axis at the origin. By applying the horizontal line test:

• For any horizontal line with \(y = k\) (where \(k > 0\)), such a line will intersect both the upward parabola from the origin and the downward parabola beyond, with two intersections occurring generally in this region.
• Similarly, a horizontal line with a negative \(y = -k\) intersects both downward branches of the graph in the negative section, indicating multiple intersections here as well.

These multiple intersections confirm that \(f(x)\) is not one-to-one, as a one-to-one function must allow any horizontal line to intersect its graph at most once. Thus, while the visual graph provides an intuitive understanding, the horizontal line test confirms the function's qualities analytically.