REPRESENTATION OF FUNCTIONS

A function may be represented by

(i) a set of ordered pairs

(ii) a table form

(iii) an arrow diagram

(iv) a graphical form

Let f : A --> B be a function.

(i)  Set of ordered pairs :

The set f = {(x, y) | y ∈ f(x), x ∈ A} of all ordered pairs represent a function.

(ii)  Table form :

The values of x and the values of their respective images under f can be given in the form of a table.

(iii)  Arrow diagram :

An arrow diagram indicates the elements of the domain of f and their respective images by means of arrows.

(iv)  A graph :

The ordered pairs in the collection

f = {(x,y) | y = f (x), x ∈ A}

are plotted as points in the xy- plane. The graph of f is the totality of all such points.

Every function can be represented by a curve in a graph. But not every curve drawn in a graph will represent a function.

Vertical line test will help us in determining whether a given curve is a function or not.

Vertical line test :

A curve drawn in a graph represents a function, if every vertical line intersects the curve in at most one point.

Example : 

Let A = {1, 2, 3, 4} and B = {2, 5, 8, 11, 14} be two sets.

Let f : A --> B be a function given by f(x) = 3x - 1. Represent this function

(i) by arrow diagram

(ii) in a table form

(iii) as a set of ordered pairs

(iv) in a graphical form

Solution :

A = {1, 2, 3, 4} and B = {2, 5, 8, 11, 14}

f(x) = 3x - 1

f(1) = 3(1) – 1 = 3 – 1 = 2

f(2) = 3(2) – 1 = 6 – 1 = 5

f(3) = 3(3) – 1 = 9 – 1 = 8

f(4) = 4(3) – 1 = 12 – 1 = 11

(i) Arrow diagram :

(ii)  Table form :

The given function f can be represented in a tabular form as given below.

(iii) Set of ordered pairs :

The function f can be represented as a set of ordered pairs as

f = {(1, 2), (2, 5), (3, 8), (4, 11)}

(iv) Graphical form :

We can plot the following points in the xy -plane to get the graphical form.

(1, 2), (2, 5), (3, 8), (4, 11)

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