Problem 115 - College Algebra

Question

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.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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.

Step 3: 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.

Step 4: 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.