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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 as rational functions can express as a ratio of two polynomials (in which the denominator is not zero). However, the reverse is true: every polynomial function is a rational function.

Step by step solution

01

Define a Rational Function

A rational function is a function which can be expressed as a fraction whose numerator and denominator are polynomials. In other words, a rational function is a ratio of two polynomial functions, where the denominator is not equal to zero.
02

Define a Polynomial Function

A polynomial is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Essentially, a polynomial is made up of one or many terms, which are not divided by variable values.
03

Compare Rational and Polynomial Functions

Though every polynomial can be viewed as a rational function, the converse is not always true. The main difference originates from the fact that the denominator of a rational function can be a polynomial, which isn't possible with polynomial functions as they only consist of numerator terms.
04

Discuss Statement When Adjectives are Reversed

If the adjectives 'rational' and 'polynomial' are reversed, the statement would be true. This is because every polynomial is a rational function where the denominator is equal to 1 or any non-zero constant.

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