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\( f(x)=\left\{\begin{array}{cc}5 & \text { if } x \leq-3 \\ -x+2 & \text { if }-3
Ask by Potter Estrada. in the United States
Mar 06,2025

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The function \( f(x) \) is defined as: - \( f(x) = 5 \) when \( x \leq -3 \) - \( f(x) = -x + 2 \) when \( -3 < x < 3 \) - \( f(x) = 2 \) when \( x \geq 3 \) For example: - \( f(-4) = 5 \) - \( f(-2) = 4 \) - \( f(0) = 2 \) - \( f(3) = 2 \) - \( f(4) = 2 \) The function is constant at 5 for \( x \leq -3 \), decreases linearly between \( -3 \) and \( 3 \), and remains constant at 2 for \( x \geq 3 \).

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This piecewise function has some interesting historical roots! Piecewise functions have been studied since the early days of calculus, where mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz explored how to describe changing relationships in a variety of contexts. The idea of using segments to describe different behaviors makes these functions vital for modeling real-world situations such as economics, physics, and engineering. In real-world applications, this function can represent a scenario where a service or product operates under different conditions. For instance, think of a delivery charge that remains constant up to a certain distance, varies as distances increase, and then reverts back to a fixed fee beyond a particular range. When interpreting such functions, it’s crucial to examine boundaries and transitions—after all, the magic happens right where the equations meet!

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