/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 115 Is every rational function a pol... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 115

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.

Short Answer

Expert verified
No, not every rational function is a polynomial function because rational functions can have negative powers or division by a variable which polynomials cannot. However, every polynomial function can be considered a rational function.

Step by step solution

01

Understanding Polynomial functions

A polynomial function is an expression consisting of variables and coefficients, which only employs the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Examples can be \(x^2\), \(3x^2 + 2x + 1\) etc.
02

Understanding Rational functions

A rational function is the ratio of two polynomials, with both polynomials in the ratio being defined by the rules of polynomial functions. Such functions look like this: \(f(x) = P(x)/Q(x)\), where both P and Q are polynomial functions.
03

Comparing the two

Every polynomial could be seen as a rational function. For example the polynomial function \(P(x) = x^2\) can also be defined as a rational function \(f(x) = x^2/1\). However, not every rational function is a polynomial. For example the rational function \(f(x) = 1/x\) is not a polynomial because, based on the definition, polynomials do not have negative powers or division by a variable.
04

Reversing the statement

If we reverse the adjectives, we say: 'Every polynomial function is a rational function'. This statement is true. Polynomials, such as \(x^2 + 2x + 1\), can be expressed as a ratio of two polynomials, such as \((x^2 + 2x + 1) / 1\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x-5}{10 x-2}+\frac{x^{2}-10 x+25}{25 x^{2}-1} $$

The equation for \(f\) is given by the simplified expression that results after performing the indicated operation. Write the equation for \(f\) and then graph the function. $$ \frac{x}{2 x+6}-\frac{9}{x^{2}-9} $$

In Exercises 100鈥�103, determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed \(f(x)=(x+2)^{3}(x-4)^{2},\) and the graph touched the \(x\) -axis and turned around at \(-2\)

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has vertical asymptotes given by x=-2 and x=2, a horizontal asymptote y=2, y -intercept at \frac{9}{2}, x -intercepts at -3 and 3, and y -axis symmetry.

Write the equation of a rational function$$ f(x)=\frac{p(x)}{q(x)} \text {having the indicated properties in which the degrees} $$ of p and q are as small as possible. More than one correct function may be possible. Graph your function using a graphing utility to verify that it has the required properties. f has no vertical, horizontal, or slant asymptotes, and no x -intercepts.
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.