# Must a rational function always be in numerator-denominator form?

This may seem trivial, but I'm looking at two examples from high school math books and wondering if they are really examples of rational functions.

The first is a line $\overline{DT}$ made up of two segments, $\overline{DS}$ and $\overline{ST}$, i.e., $\overline{DS} + \overline{ST} = \overline{DT}$. $\overline{DS}$ is associated with the expression $2x-8$ and $\overline{ST}$ with $3x-12$. It seems like this is a ratio issue, thus, some sort of rational function?

The second is from a high school Algebra 2 text's section specifically on rational functions.

A cylinder has a volume of $(x+3)(x^2-3x-18)\pi$ cubic centimeters. Find the height of the cylinder.

Good, I know how to "do" it, but, again, this doesn't seem to be in the classic numerator-over-denominator form of a proper rational function. Why is this a rational function, and, was the first also a rational function of some sort?

• $(x+3)(x^2+3x-18)\pi=\frac{(x+3)(x^2+3x-18)\pi}{1}$
– J.R.
May 7, 2016 at 17:49
• It is usual to allow the denominator to be 1 in a rational function. May 7, 2016 at 17:50
• A rational function is essentially anything you can get by adding x into the real numbers and using the four arithmetic operations. Your functions fit this format, so they are clearly rational functions. (If you absolutely want them to have denominators, imagine them as having a denominator of 1.) May 7, 2016 at 17:51
• Yes and this is called a polynomial. Polynomials are rational functions.
– J.R.
May 7, 2016 at 17:51
• Consider the following polynomial $P(x) = 1$ ... May 7, 2016 at 21:03