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Piecewise functions are mathematical expressions that have different values for different intervals of the input variable, typically defined using multiple sub-functions. Each sub-function applies to a specific interval and is usually accompanied by a set of conditions that determine when it should be used.

In the case of the function \(f(x) = x|x|\), we have a textbook example of a piecewise function because it behaves differently on different intervals of \(x\). When \(x \geq 0\), the absolute value of \(x\) remains \(x\), turning the function into \(f(x) = x^2\). Conversely, when \(x < 0\), the absolute value of \(x\) becomes \(-x\), and the function transforms into \(f(x) = -x^2\). Understanding how to work with piecewise functions is crucial in calculus, as they can model a variety of real-world scenarios with distinct behaviors over different domains.

The derivative of an absolute value function requires understanding that the absolute value creates a 'kink' in the graph, where the function transitions from one piece to another. Due to this, finding the derivative involves considering the separate pieces of the function.

Using the function from the exercise, \(f(x) = x|x|\), we observe that it is composed of two differentiable functions, \(f(x) = x^2\) for \(x \geq 0\) and \(f(x) = -x^2\) for \(x < 0\). The respective derivatives are \(f'(x) = 2x\) for \(x \geq 0\) and \(f'(x) = -2x\) for \(x < 0\). For points where the function is not differentiable -- in this case, at \(x=0\) due to the absolute value condition -- one usually examines the behavior of the derivative from the left and the right side of the point to understand the function's behavior at that specific point.

Sketching parabolas is a fundamental skill in graphing functions, particularly those of the form \(y = ax^2 + bx + c\). These quadratic functions create U-shaped curves that either open upwards or downwards depending on the sign of the coefficient \(a\).

To sketch a parabola, it's important to determine its vertex, axis of symmetry, and general shape (whether it opens up or down). For the given exercise, the graph consists of two parabolas with a common vertex at \((0,0)\). For \(x \geq 0\), the function \(f(x) = x^2\) is represented by a parabola that opens upwards, while for \(x < 0\), the function \(f(x) = -x^2\) sketches a parabola that opens downwards. By correctly plotting these two halves, students can visualize and better understand the behavior of the entire function.

Symmetry in functions is a property that allows for easier understanding and prediction of a function's behavior. A function can be symmetric about the y-axis (even symmetry), x-axis, the origin (odd symmetry), or none. For a function to be symmetrical about the y-axis, it must satisfy the condition \(f(-x) = f(x)\).

Our exercise's function exhibits even symmetry since for every \(x\), the negative counterpart \(-x\) yields the same value of the function when the absolute value is considered. This means that one-half of the function is a mirror image of the other half across the y-axis. Understanding symmetry can simplify graphing and analyzing functions because once one-half of the graph is known, the other half is readily determined by reflecting across the line of symmetry.