Question

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

Answer 1

Not every rational function is a polynomial function because rational functions include division by a polynomial and may not be defined at all points. However, every polynomial function is a rational function because a polynomial divided by one is still a polynomial.

Step-by-step explanation:

In the context of mathematical functions within economic models, we can discuss the nature of polynomial functions and rational functions. A rational function is a ratio of two polynomials, where the denominator is not zero. In contrast, a polynomial function is a sum of terms, each consisting of a variable raised to a non-negative integer power and multiplied by a coefficient.

Now, addressing the question: Is every rational function a polynomial function? The answer is no. Every rational function is not a polynomial function because it includes a division by another polynomial, and it can have points (like an asymptote) where the function is not defined.

Is every polynomial function a rational function? The answer is yes. Every polynomial function can be considered a rational function with the denominator set to one, which will not affect the definition of the polynomial function since dividing by one does not change the value of the expression.

Answer 2

No, not every rational function is a polynomial function, but every polynomial function is a rational function.

Step-by-step explanation:

Rational functions: These are functions expressed as the ratio of two polynomials. Examples include 1/(x^2 + 1) and (x - 3)/(x + 1).

Polynomial functions: These are functions where the highest power of the variable has a non-negative integer exponent, and the function only involves addition, subtraction, multiplication, and these exponents. Examples include x^2 + 3x - 1 and 5x^3 + 2x.

Relationship:

Every polynomial function can be written as the ratio of itself and 1 (a constant polynomial), making it a rational function. For example, x^2 + 3x - 1 can be expressed as (x^2 + 3x - 1) / 1.

However, not every rational function can be expressed as a polynomial. Functions like 1/(x^2 + 1) have non-integer exponents in their denominators, making them non-polynomial.

Therefore, while all polynomial functions are also rational functions, the reverse is not true.