## Rational Functions

Rational functions are basically a division of a polynomial function with another polynomial function.

f(x)= s(x)/ r(x)

Where f(x), s(x) and r(x) are polynomial functions, r(x) cannot be zero and the degree of r(x) > 0.

The simplest rational function is of the form f(x) = y = k/x where k is a constant. It is also called the inverse
variation function
because y varies inversely with x which means that as the value of x increases, that of y
decreases.

Example

xy = k can be written as y = k/x and hence represents an inverse relation between x and y.

Example

The rate of flow of water through a pipe is x cm3 per second. Express the time required to fill a tank of volume k
cm3 as a function of x.

Solution

We can solve this problem by using the unitary method.

The time required to fill x cm3 of water in the tank = 1 second.

• Time required to fill 1 cm3 of water in the tank = 1/x second.

• t = time required to fill k cm3 of water in the tank = k/x second.

t = k/x; therefore, time varies inversely with x.

You will encounter rational expressions in many problems related to the ratio of two quantities.

#### Try this problem

• Find the ratio of the area of rectangle A with dimensions 4x and 4x+8 to Rectangle B with dimensions 2x and 2x +2.

Area of rectangle A ÷ Area of rectangle B = 