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Piecewise functions are mathematical expressions defined by multiple sub-functions, each applying to a certain interval of the main function's domain. In the case of our example function, there are two equations. For the interval from 0 to just before 3, the function is given by the rule \(f(x) = x + x^2\). For values of \(x\) greater than 3 and up to 5, the function rule changes to \(f(x) = 2x\).

Understanding piecewise functions involves recognizing the intervals and applying the correct function rule to each piece. This helps students analyze complex functions that may behave differently over various intervals. Such functions are particularly useful in representing real-world scenarios where a relationship changes based on different conditions or domains.

The idea of symmetry in functions concerns how a function behaves when its inputs are reversed or negated. There are two primary types of symmetry in functions: even and odd.

An even function fulfills the condition \(f(x) = f(-x)\), meaning that the function's graph is symmetrical with respect to the y-axis. Conversely, an odd function meets the condition \(f(-x) = -f(x)\), indicating that the graph is symmetrical with respect to the origin.

In the step-by-step solution, the even extension was found by creating a mirror image of the function across the y-axis, while the odd extension was found by rotating the function 180 degrees about the origin. This concept helps students to extend the function beyond its original domain and also paves the way for further exploration in Fourier series, where functions are expressed in terms of their even and odd components.

Extending a function refers to the process of expanding its domain while maintaining the original properties or patterns of the function. In our step-by-step solution, the extension process uses the function's existing behavior to predict how it would act outside its original domain. For an even function, this involves creating a symmetric extension on the negative side of the x-axis. For an odd function, it involves extending the function such that the pattern \(f(-x) = -f(x)\) holds.

Extending a function like this can reveal how it will behave in new situations, predict the values outside the known range, and provide insight into the function's potential for modeling phenomena across a broader scope. Understanding how to extend functions correctly is essential for students tackling real-world problems where the behavior of a system needs to be predicted beyond the observed data.