A function f : X � Y
is a rule which associates each element x �
X with a unique element y �
Y such that y = f(x).
The set X is called the domain
of f and is denoted by D_{f} .
The set Y is called the codomain
of f.
f(x) is called the image of x
under f or f-image of x.
The set { f(x) : x �
X } is called the range
of f and is denoted by R_{f} .� It is the set
of all images of x under f.
Range of f is a subset of Y, ie, R_{f}
� Y.
From the graph of f, f is a function
if every vertical line x = a for each a �
X cuts the graph of f at only one point.
Note:
In defining a function, the rule and domain must be given.�
If the domain is not given, it is taken to be the largest possible
domain for which the function is defined.�
If the codomain is not specified, it is taken to be the range of the
function.
njective, Surjective, Bijective
A function f : X � Y
is said to be injective or one-one
if no two distinct elements of X have the same f-image,
A function f : X �
Y is said to be surjective or onto if for
every y � Y, there is at
least one x � X such that
y = f(x),
ie, R_{f} = codomain of f.
A function f : X �
Y is said to be bijective
if it is both one-one and onto.
nverse Function
For every bijective function f : X
� Y, there exists an inverse function
f^{-1} : Y �
X such that
y = f(x)� ��
x = f^{-1}(y).
Domain of f^{-1}
= Range of f.
Range of f^{-1}
= Domain of f.
The graph of the inverse y = f^{-1}(x)
is the reflection of the graph of y = f(x) in the
line y = x.� Same scale must be used for both axes.
omposite Function
Let f and g be functions.
Then the composite gof,
or simple gf, is defined by
gof(x) = g(f(x)).
gof is a function
if R_{f} � D_{g}.
D_{gof}
= D_{f }.
R_{gof}
� R_{g}.
xamples
Example 1: Two
functions are defined as follows:
f : x �
x^{2} - 2x,
x �
R, x � 0;
g : x �
e^{2x},
x �
R.
For each of the functions, state the range and determine
whether or not the function is one-one.
Give, in the same form, the definition of the
functions gof and g^{-1}.
Solution:
x^{2} - 2x
�=�
x(x - 2)
�=�
(x - 1)^{2} -
1.
R_{f} = [-1, �).�
f(0) = 0 = f(2)� but� 0 �
2.
\� f is not one-one.
R_{g} = R^{+} = (0, �).�
g is one-one,�
as every horizontal line y = b, b � R,
cuts the graph of y = g(x) at most once.
gf(x)
�=�
g(x^{2} - 2x)
�
�=�
2x^{2} - 4x
e
\� gof
: x �
2x^{2} - 4x
e
, x � R, x �
0.
Let y
�=�
e^{2x}
ln y
�=
2x
x
�=�
�ln y
\� g^{-1}
: x � �ln x, x
� R^{+}.
Example 2: The
functions f and g are defined by
f : x �
ln x,
�x > 0;
g : x �
1 - x,
�x < 1.
Show that fog
is a function.� Define fog and
state its range.� Explain why the function gof
does not exist.
Solution:
R_{g} = R^{+}, D_{f} = R^{+},
\� R_{g} �
D_{f}� �� fog
is a function.
fog(x)
�=�
f(1 - x)
�=�
ln (1 - x)
\� fog
: x � ln (1 -
x), x < 1.
R_{fog} = R.
�
R_{f} = R, D_{g} = (-�,
1),
\� R_{f} �
D_{g}� �� gof
is not a function.