Rational vs. Polynomial Functions

[Lizardjuice7]

In my calc class we are reviewing rational and polynomial functions before we start with the actual calculus part of the course.

In my book we had 3 problems that we had to do for homework and none of my classmates could understand why the book answered them a certain way.

Question:

State whether the functions below are Rational, Polynomial, Neither, or Both.

1. f(x)=3x²+2x+1

2. f(x)=(2x²+4x)/(x-1)

3. f(x)=(x-1)^1/2From what I understand a rational function needs to have a fraction bar in it, so number two is rational. Number 3 is raised to the 1/2 power so it is neither. Not rational nor a polynomial. I thought that number one would just be a polynomial, but my teacher's book says it is both. We discussed it in class and couldn't make sense of it.

If someone could explain to me what the functions above are that would be very helpful.

Thanks!

 

[jeffreydk]

The definition of a rational function is any function that satisfies/can be expressed as

$$f(x)=\frac{P(x)}{Q(x)}$$

So when thinking whether (3) is rational or not, think about what happens if $Q(x)=1$ in this definition.

 

[symbolipoint]

Lizardjuice7, your thinking is fine. Actually a book I have states that monomials can be polynomials (seems strange), giving credibility to jeffreydk's justification.

 

[Lizardjuice7]

Thanks,

That really helps a lot!

Lizardjuice7