How do you find the domain and range of a rational function(or any other function) using Calculus

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I know that the only ways to find the range of a rational function is using an inverse, graphing, or using calculus. How would you use calculus to find the range of a rational function (or any other type of function) using Calculus?

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You can find its limiting behaviour as $x \to \infty$ and $x \to -\infty$, and also find local maxima and local minima. Don't forget to account for vertical asymptotes that arise when the denominator is $0$.