Functions and it's application Notes

Function 

A function is a special type ofrelation that expresses how one quantity depends on another quantity. For example, when money is invested at some interest rate, the interest 'I' depends on the length of time,  this dependence can be expressed by saying that "I is a function of "t". 

Let A and 8 be two non-empty sets. A function /from A to 8 is a rule which associates to each element x of A a unique y of8. The element y is called the image of x under f and we write f(x) = y  

which is read "/of x equals y". A is called the domain off and B is called the co-domain off The set of all images of the elements of the domain is called the range off The elements x and y are called the  

independent and dependent variables respectively.

Functions related to Business and Economics.

 In this section we present some functions which are very useful in business and economics. 

Demand Function. An equation that relates price per unit and quantity demanded at that price is called a demand function. Its graph is called a demand curve. If p is the price per unit of a certain product and x is the number of units of that product that 

consumers will demand during some time

period at that price, then we can express

the demand function as X = f(p) 

Here x is the dependent variable

and p is the independent variable. Since

negative prices and quantities are

meaningless, both x and p must be non-

negative. Usually, an increase in price

corresponds to a decrease in quantity

demanded and vice versa. demand curve which is a straight line, it is  called a linear demand curve. , the horizontal  axis is used for the dependent variable x and the vertical axis for the independent variable p.

Supply Function.

 An equation that relates price per unit and the quantity supplied at that price is called a supply function and its graph called a supply curve. If p denotes the price per unit and x denotes the corresponding quantity supplied, then the supply function can be expressed as  X = g {p)  

As before x and p are non-negative. Usually, an increase in price corresponds to an increase in the quantity supplied and a decrease in the price brings about a decrease in supply.

Cost Function.

 Let C the total cost incurred in the production of x units of a commodity. Then a function, say, C = C (x)  relating C and x is called a cost function and its graph is called a cost curve. It may be noted that 

 total cost = fixed cost + variable cost,  

where fixed cost (or overhead) is the sum of all costs that are independent of the level of  production, such as rent,insurance etc. and variable cost is the sum of al I costs that are dependent  on the level of production, such as cost of  material, labour, etc.  

Revenue Function. 

Let R be the total revenue or income the company makes by selling x units of a

product at price p per unit. Then R is given by the fomrnla    R =px  

R is called the total revenue function. 

Profit Function.

 If R(x) and C(x) be the total revenue reveived and the total cost incurred, in the

production ofx units of a product, then the function P given by

  P (x) = R (x) - C (x) 

 is called the profict function

.  

Break even point. 

The break-even point is the level of production where the revenue from the sales is equal to the cost of production. At the break even point, the company is neither making a profit nor  losing money.  

Consumption Function.

 If I denotes the total national income and C denotes the total national consumption, then the function  

C = f(l)

relating I and C is called the consumption function. The difference between / and consumption C is savings S. Thus S = I - C.

Questions :- 1

A publishing house finds that the cost of production directly attributed to each book is Rs. 40 and that the fixed costs are Rs. 25,000. If each book can be sold for Rs. 60, then determine:  

(i) the cost function,

(ii) the revenue function,

(iii) the profit function, and

(iv) the break-even point.

Questions:- 2

A profit making company wants to launch a new product. It observes that the fixed cost of the new product is Rs. 35,000 and the variable cost per unit is Rs. 500. The revenue function for the  sale of x units is given by 5000x - 100x".

 Find (i) profit function, (ii) break-even values, and (iii) the  values of x that result in a loss.

*(" It's mean square of x)

The Next topic is Limit and it's application of business mathematics.

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