Suppose we know that the cost of making a product is dependent on the number of items, x, produced. This is given by the equation [latex]C\left(x\right)=15,000x - 0.1{x}^{2}+1000[/latex]. If we want to know the average cost for producing x items, we would divide the cost function by the number of items, x.

The average cost function, which yields the average cost per item for x items produced, is

[latex]f\left(x\right)=\frac{15,000x - 0.1{x}^{2}+1000}{x}[/latex]

Many other application problems require finding an average value in a similar way, giving us variables in the denominator. Written without a variable in the denominator, this function will contain a negative integer power.

In the last few sections, we have worked with polynomial functions, which are functions with non-negative integers for exponents. In this section, we explore rational functions, which have variables in the denominator.