A rational function is a function that can be expressed as the ratio of two polynomial functions. It is a mathematical expression that can be used to model a wide range of real-world phenomena, from population growth to the behavior of electrical circuits.
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Rational functions can be used to model a wide range of real-world phenomena, including population growth, the behavior of electrical circuits, and the motion of objects under the influence of gravity.
The domain of a rational function is the set of all real numbers except for the values of the independent variable that make the denominator equal to zero, which are called the vertical asymptotes.
Rational functions can have both horizontal and vertical asymptotes, which can be used to analyze the behavior of the function as the independent variable approaches positive or negative infinity.
The derivative of a rational function is itself a rational function, which can be used to analyze the rate of change of the original function.
L'Hôpital's rule is a powerful tool for evaluating the limit of a rational function when the numerator and denominator both approach zero or infinity.
Review Questions
Explain how the concept of a rational function is related to the review of functions in section 1.1.
Rational functions are a specific type of function that can be expressed as the ratio of two polynomial functions. In the review of functions in section 1.1, the general properties and characteristics of functions, such as domain, range, and graphical behavior, are discussed. These concepts are directly applicable to rational functions, as they are a specific class of functions that exhibit these properties and can be analyzed using the techniques covered in the review of functions.
Describe how the differentiation rules covered in section 3.3 can be applied to rational functions.
The differentiation rules covered in section 3.3, such as the power rule, the product rule, and the quotient rule, can be used to differentiate rational functions. Specifically, the quotient rule, which states that the derivative of a fraction is the derivative of the numerator times the denominator minus the numerator times the derivative of the denominator, all divided by the square of the denominator, is particularly useful for differentiating rational functions. This allows for the efficient calculation of the derivative of a rational function, which can be used to analyze its behavior and properties.
Explain how L'Hôpital's rule, covered in section 4.8, can be applied to evaluate the limits of rational functions.
L'Hôpital's rule, discussed in section 4.8, is a powerful tool for evaluating the limits of rational functions when the numerator and denominator both approach zero or infinity. This rule states that if the limit of a fraction is in the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then the limit is equal to the limit of the derivative of the numerator divided by the derivative of the denominator. This allows for the efficient evaluation of limits involving rational functions, which can be crucial in understanding their behavior and properties.
A polynomial function is a function that can be expressed as the sum of a finite number of non-negative integer powers of the independent variable, with coefficients that are real numbers.
Asymptote: An asymptote is a line that a graph of a function approaches but never touches. Rational functions can have both horizontal and vertical asymptotes.
The inverse function of a rational function is another rational function that, when composed with the original function, results in the identity function.