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Functions and Statements

Functions in economics The concept of function has special significance in economics. Generally, the various laws and theories of economics are expressed in the form of functions. Some of the important functional relationships are: demand function, supply function, cost function, etc. The above figure shows the graphs of demand and supply functions in economics.

Functions

Definition of a function: Let A and B be two non-empty sets. A function 'f' from A to B, written as f : A B, is a relation in which every element 'x' of A corresponds to only one element f(x) of B. The element f(x) is called the image of 'x' and the element 'x' is called the pre-image of f(x). The ordered pairs of a function 'f' are represented as (x, f(x)).

Domain, co-domain and range of a function: Let 'f' be a function from non-empty set A to another non-empty set B. Then, the non-empty set A is called the ‘domain’ of the function 'f', the non-empty set B is called the ‘co-domain’ of the function 'f' and {f(x): x A} B is called the ‘range’ of the function. In some cases, the range happens to be identical with the co–domain of a function.

  • Ex: Let A = { – 1, 1, 2, 3, 4} and B = {1, 4, 9, 16, 24} and the rule f(x) = x2, x A is a function from A to B then find domain, co–domain and range of a function.

    Sol: Given, f: A B is a function. Using the rule f(x) = x2, x A we have:
    f(–1) = (–1)2 = 1; f(1) = (1)2 = 1; f(2) = (2)2 = 4; f(3) = (3)2 = 9; f(4) = (4)2 = 16
    ∴ Domain of a function = A = {– 1, 1, 2, 3, 4};
        Co-domain of a function = B = {1, 4, 9, 16, 24};
        Range of a function = {1, 4, 9, 16}.
Representation of a Function

There are six ways to represent a function. They are: (i) roster form, (ii) set builder form, (iii) by formula, (iv) by table, (v) by arrow diagram and (vi) by graph.

  • Roster form: In this form, the function is represented by the set of all ordered pairs which belong to the function. For example, let A = {3, 6, 9}, B = {1, 2, 3} and f be the function "is thrice of" from A to B. Then, f = {(3, 1), (6, 2), (9, 3)}.
  • Set builder form: In this form, the function is represented as: f = {(x, f(x)) : f(x) = x + 2}. For example, let f = {(3, 2), (6, 5), (9, 8)}. Then, set builder form of a function f is: f = {(x, f(x)) : f(x) = x – 1}.
  • By formula: In this form, a formula, i.e., an algebraic equation can be used to represent a function. For example, an equation f(x) = x + 4 represents a function where x takes all values on set of natural numbers (N) and the values of f(x) are obtained by using the above function.
  • By table: In this form, a table can be used to represent a function. For example, the table given below represents a function: f(x) = x + 4.
  • x 1 2 3 4 5
    f(x) 5 6 7 8 9
  • By arrow diagram: In this form, the function is represented by drawing arrows from the first elements to the second elements of all ordered pairs which belong to the given function.
  • By graph: In this form, the function is represented by drawing dots in the graph for all ordered pairs which satisfy the given function.
Types of Functions

Many-one Function

A function f : A B is said to be many-one, if two or more than two elements in A have the same image in B, i.e., f : A B is a many-one function, if there exists x, y A such that x ≠ y but f(x) = f(y).

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