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First, let’s dive into the historical significance of rational functions. They have been a cornerstone of mathematical studies since the time of ancient civilizations, who used them for various practical applications, from land measurement to trade. The study of these functions can be traced back to the likes of Euclid and later mathematicians, who began to formalize their properties in relation to algebra and geometry, paving the way for what we now consider calculus and advanced mathematics.

Now, let’s talk about the real-world application of rational functions! They appear in numerous scenarios, from economics—like calculating cost functions or profit margins—to physics, where they can model the behavior of physical systems like resistance in electrical circuits. Understanding how to manipulate and visualize these functions is crucial for engineers and scientists, who often rely on these mathematical tools to predict outcomes and design efficient solutions!