Rational Functions

A rational function is the ratio of two polynomials, where the denominator is not equal to zero.

Rational functions help model real-world costs. For example, consider a small company that produces reusable water bottles with a fixed cost of $2,000 and a production cost of $8 per bottle.

If x bottles are produced, the total cost equals the fixed cost plus x times the cost per bottle. To find the average cost per bottle, we divide the total cost by the number of bottles produced, giving us a rational function.

Using this function, for 100 bottles, the average cost per bottle is $28. For 500 bottles, it drops to $12. Because fixed costs are spread across more units, producing more bottles lowers the average cost per bottle.

Without production, an average cost cannot be calculated. This excludes x = 0, giving a domain of positive integers for bottle production.

In this function, 2000/x is always positive for positive x, the average cost is always more than 8.

As production grows, 2000 over x becomes smaller, so the average cost approaches $8 but never equals it. This gives a range strictly greater than $8.