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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, every rational function is not a polynomial function. However, if we reverse the adjectives, the statement that every polynomial function is a rational function is true.

Step by step solution

01

- Definition of Polynomial Function

A polynomial function is a function such as \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_2 x^2 + a_1 x + a_0,\] where \(a_n, a_{n-1}, \ldots, a_2, a_1, a_0\) are constants, and \(n\) is a non-negative integer.
02

- Definition of Rational Function

A rational function is any function which can be defined by a rational fraction. In simpler terms, a rational function is one that is the division of two polynomial functions. It looks like \[f(x) = \frac{P(x)}{Q(x)}\] where \(P(x)\) and \(Q(x)\) are polynomial functions.
03

- Comparison

Every polynomial function can be considered as a rational function where the denominator is 1. However, not all rational functions are polynomial, because rational functions have denominators other than 1, and the denominator can be a variety of polynomial functions.
04

- Answering the second part of the Question

If we reverse the adjectives, stating that every polynomial function is a rational function, we get a true statement because a polynomial function could be expressed as a ratio of itself to 1, hence it's a rational function.

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