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Chapter 3: Problem 76

What is a rational function?

Short Answer

Expert verified
A rational function is a function that can be defined by a ratio of two polynomials. For instance, \(f(x) = \frac{x^2 + 3x + 2}{x - 1}\) is a rational function.

Step by step solution

01

Define Rational Function

A rational function is any function that can be defined by a rational fraction, i.e., an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers, they could be taken in any field K. In this context, a field K is a set equipped with two operations, called addition and multiplication.
02

Explain division by zero

A rational function is not defined when the denominator is zero. When the denominator of a fraction is 0, the fraction represents division by zero, and the value of the fraction is undefined.
03

Provide example of rational function

Here's an example of a rational function: \(f(x) = \frac{x^2 + 3x + 2}{x - 1}\) . Note, it's in the form of a ratio of two polynomials, and is not defined when \(x = 1\) because that would make the denominator zero.

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Most popular questions from this chapter

A herd of 100 elk is introduced to a small island. The number of elk, \(N(t),\) after \(t\) years is described by the polynomial function \(N(t)=-t^{4}+21 t^{2}+100\) a. Use the Leading Coefficient Test to determine the graphs end behavior to the right. What does this mean about what will eventually happen to the elk population? b. Graph the function. c. Graph only the portion of the function that serves as a realistic model for the elk population over time. When does the population become extinct?

In Exercises \(21-26,\) use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. $$f(x)=11 x^{4}-6 x^{2}+x+3$$

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=x^{4}-2 x^{3}+x^{2}$$

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

In Exercises \(74-77\), use a graphing utility with a viewing rectangle large enough to show end behavior to graph each polynomial function. $$f(x)=-2 x^{3}+6 x^{2}+3 x-1$$
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