A function may be represented by
(a) a set of ordered pairs
(b) a table form
(c) an arrow diagram
(d) a graphical form

**Representation of Functions**

A function may be
represented by

(a) a set of
ordered pairs

(b) a table form

(c) an arrow diagram

(d) a graphical form

Let *f*: *A*→
*B* be a function

**(a) Set of ordered pairs**

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

**(b) Table form**

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

**(c) Arrow diagram**

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

**(d) 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.

The following test
will help us in determining whether a given curve is a function or not.

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

Using vertical line
test, determine which of the following curves (Fig.1.18(a), 1.18(b), 1.18(c),
1.18(d)) represent a function?

The curves in
Fig.1.18(*a*) and
Fig.1.18(*c*) do not represent a function as
the vertical lines meet the curves in two points *P *and *Q.*

The curves in
Fig.1.18(b) and Fig.1.18(d) represent a function as the vertical lines meet the
curve in at most one point.

Any equation
represented in a graph is usually called a ‘curve’.

**Example 1.11** 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*) = 3*x *− 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

*A *= {1, 2, 3, 4} ; *B *=
{2, 5, 8,11,14} ; *f *(*x*) = 3*x *– 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**

Let us represent the function* *: *A *→ *B *by an
arrow diagram (Fig.1.19).

**(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**

In the adjacent *xy *-plane
the points

(1,2), (2,5), (3,8),
(4,11) are plotted (Fig.1.20).

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