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Chapter 3: Problem 108

What is a rational function?

Short Answer

Expert verified
A rational function is a mathematical function that can be expressed as the quotient of two polynomial functions.

Step by step solution

01

Definition of a Function

In math, a function is a rule that relates an input to an output. It means for each input, you will get one output and this mapping of input to output should be clearly specified.
02

Introduction to Polynomial Functions

A polynomial function is a function that can be expressed in the form \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0 \), where \( n \) is a nonnegative integer and the \( a_i \)'s are the coefficients.
03

Definition of a Rational Function

A rational function is a function of the form \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomial functions. Rational functions are versatile and can express a wide variety of mathematical situations. It's important to note that, in a rational function, the denominator \( Q(x) \) cannot be zero because division by zero is undefined.

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Most popular questions from this chapter

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{3 x-7}{x-2}$$

Exercises will help you prepare for the material covered in the next section. Factor: \(x^{3}+3 x^{2}-x-3\)

A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?

Find the slant asymptote of the graph of each rational function and b. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}+1}{x}$$
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