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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?

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    $\begingroup$ $(x+3)(x^2+3x-18)\pi=\frac{(x+3)(x^2+3x-18)\pi}{1}$ $\endgroup$
    – J.R.
    May 7, 2016 at 17:49
  • $\begingroup$ It is usual to allow the denominator to be 1 in a rational function. $\endgroup$
    – almagest
    May 7, 2016 at 17:50
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    $\begingroup$ 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.) $\endgroup$
    – Ege Erdil
    May 7, 2016 at 17:51
  • $\begingroup$ Yes and this is called a polynomial. Polynomials are rational functions. $\endgroup$
    – J.R.
    May 7, 2016 at 17:51
  • $\begingroup$ Consider the following polynomial $P(x) = 1$ ... $\endgroup$
    – John Joy
    May 7, 2016 at 21:03

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Contrary to a previous comment, the system of polynomials are like integers: they are not closed under division. This means that rational functions are a superset of polynomials, like rational numbers are a superset of the integers. That is why most text books devote separate units to rational functions.

Returning to the questions at hand. The intent of the geometry question is unclear. Are students using ratios to solve the problem? Or is the sum of the segments given, and students should find the value of x?

For the cylinder problem, I think this is a good exercise in a unit on rational functions because the solution involves creating a rational function. If I gave you the volume of a cylinder in numeric form, you could find the height by dividing. Here, the volume of the cylinder is a polynomial; you have to divide the volume by the cylinder, represented by a polynomial, by the area of its base, also represented by a polynomial. It would seem therefore, the height of the cylinder is represented by a rational function.

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