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Chapter 2: Problem 84

Is every rational function a polynomial function? Is every polynomial function a rational function? Explain.

Short Answer

Expert verified
Every rational function is not a polynomial function because the denominator in a rational function can contain terms that are not constants, which would not fit the definition of a polynomial function. However, every polynomial function is a rational function because it fits the definition of a rational function if you let the denominator \( q(x) \) be 1.

Step by step solution

01

Define 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 and \( q(x) \) is not equal to zero.
02

Define Polynomial Function

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 the coefficients (\( a_n, a_{n-1}, ..., a_0 \)) are real numbers, the exponents (\( n, n-1, ..., 0 \)) are non-negative integers, and \( x \) is the variable.
03

Analyze whether every rational function is a polynomial function

Not every rational function is a polynomial function. This is because the denominator of a rational function can contain terms that are not constants, whereas a polynomial function only includes powers of \( x \) multiplied by constants.
04

Analyze whether every polynomial function is a rational function

Yes, every polynomial function is a rational function. This is because a polynomial function fits the definition of a rational function with \( q(x) = 1 \).

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