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Chapter 2: Problem 119

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 because rational functions can have variables in the denominator. However, every polynomial function can be considered a rational function because we can express it as a ratio of two polynomials.

Step by step solution

01

Understand the Definitions

Rational functions are defined as the ratio of two polynomials. So, a function \( f(x) \) is a rational function if it can be expressed as \( f(x) = P(x) / Q(x) \) where \( P(x) \) and \( Q(x) \) are polynomial functions and \( Q(x) \neq 0 \). Polynomial functions are expressions consisting of variables and coefficients which only employ the operations of addition, subtraction, multiplication, and non-negative integers as exponents.
02

Analyze if every rational is a Polynomial

Recall that a polynomial function does not include division by a variable. So, if a rational function has a variable in the denominator, \( Q(x) \), it cannot be a polynomial function. Hence, not every rational function is a polynomial function.
03

Analyze if every Polynomial is a Rational

Every polynomial function can be considered as a rational function. This is because a polynomial \( P(x) \) can be expressed as a ratio \( P(x) / 1 \). Hence the denominator \( Q(x) = 1 \) is a polynomial. Thus, every polynomial function is a rational function.
04

Conclusion

So in conclusion, not every rational function is a polynomial function, but every polynomial function is a rational function.

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Most popular questions from this chapter

Solve each inequality using a graphing utility. $$\frac{1}{x+1} \leq \frac{2}{x+4}$$

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