# Composite Functions

An example of real-life situation that applies composite function.

A composite function represents an application of one function to the result of another. The term “composition of functions” or composite function refers to the combining of functions in a manner where the output of one function becomes the input of the next function. In mathematical terms, the range (the y – value) terms of one function becomes the domain (x – value) of the next function

The mathematical notation used for composite function is as follows.

(f O g) (x) = f (g(x)). It is read as f composed with g of x. Composite functions can be put to use for different purposes in real-world situations. For example, here the usage of composite function in a chicken farm business is used as an example.

### Problem

After graduating from the university, we plan to make a little farm in the countryside. We initially bought 300 chickens to start with our poultry business. We found out that each chicken is able to produce 1 egg each day. But, as we also want to raise the chicken for their meat, we saved 5 eggs per day and placed them in an incubator so that they can be raised until they become adult chickens. We plan to sell them in the near future for their meat. So, the number of eggs that can be sold on each day is : E (d) = (300-5)d = 295d

We sell the eggs produced each day to a distributor. Each egg has a price tag of RM 0.35. The distributor can gain a discount rate of 2% which will be accumulated and compounded every 50 eggs. Therefore, the price of the egg can be represented as

P ( E ) = 0.35E[1- {(1.02)(E/50) %}]. We made an account to keep track of our sales in the chicken farm business. Combining the two factors of our income, we made a general formula to calculate our source of income in days (d).

### Working

1st Factor – Number of eggs, E that can be sold per day (d).

E (d) = (300-5)d = 295d

2nd Factor – Price, P of an egg, E sold to the distributor.

P ( E ) = 0.35E[1- {(1.02)(E/50) %}]

Hence, the source of income in daily (d) basis can be represented as

P{E(d)} = 103.25d[ 1 - {(1.02)(295d/50)}%]

### Sample Calculation

- Income for 1 day, d = 1,

P{E(1)} = 103.25(1)[ 1 - {(1.02)(295(1)/50)}%]

Income = RM 102.09

- Income for 2 days, d = 2,

P{E(2)} = 103.25(2)[ 1 - {(1.02)(295(2)/50)}%]

Income = RM 203.89

- Income for 30 days, d = 30,

P{E(30)} = 103.25(30)[ 1 - {(1.02)(295(30)/50)}%]

Income = RM 2066.54

In the example above, the number of eggs that can be sold for a day is related to how much income that can be made. It is also related to the amount of compounded discount that a distributor can benefit. This co-relationship of the variable, number of eggs (E) is used to form a composite function to calculate the total income that can be made in a daily basis. When the number of days is 1, then the income for that one day is RM 102.09 which is after the discounted price. When the number of days is 30, then the total income for thirty consecutive days is RM 2066.54. The income decreases in daily basis as a result of the compounded discount per 50 eggs that is given to the distributor. At one point there will be a time when there is no gross income because of the accumulated discount that is given away. Therefore this function is usually used for each month and is renewed as the number of adult chicken that can produce egg might increase. From this example, it is clear how a composite function can be used in real-world situations.

Composite functions are widely used in everyday life and are more common than you may realize. For instance, a combined operation of an addition and multiplication using a calculator indicates a very simple form of composite function application. Other than that, it is very important in business and commercial sectors where composite functions helps to relate different variables to calculate income and gross profit.

Liked it

On January 14, 2009 at 1:29 pm

A very interesting article.

On January 14, 2009 at 10:48 pm

Amazing how math can be applied in a simple everyday life activity. Exellent work!

On January 22, 2009 at 12:40 am

well written