BY
Sampath
Ix class
a
g
gof
1
2
3
+
f
Georg Ferdinand
Ludwig Philipp Cantor
Born: 3 March 1845
in St Petersburg,
Russia Died:
6 Jan 1918
in Halle, Germany
GEORG CANTOR born in St petersburg, RUSSIA
TYPES OF FUNCTION








One to One Function
On to Function
One to One On to
Inverse of a Function
Equal Function
Identity Function
Constant Function
Composite Function
O N E TO
O N E
 A function f : A
B is said to be
One to One Function. If no two distinct
elements of A have the same image in B.
f
a
b
x
y
c
z
A
B
On to Function
f :A
B is said to be an On to Function. If f(A) is the image of A
equal B that is f is On to Function if every element of B. The Co-domain
is the image of at least one element A the domain.
f: A
B is on to
for every x € B there exist at least one
x € A such that f(x) = y
f(A) = B.
A
a
f
B
x
b
y
c
d
z
One to one on to
 A function f : A
B is said to be a
bijection if it is both one to one and on to.
A
X
B
f
a
Y
b
Z
c
INVERSE FUNCTION

If f is a function then the set of ordered
pairs obtained by interchanging the first
and second coordinates of each order fair
in F s called inverse of F. it denoted by F-1
f = { (0,0), (1,1), (2,4), (2,9)……..}
f -1 = { (0,0), (1,1), (4,2), (9,2)……..}
A
f
B
1
X
2
Y
3
z
IDENTITY FUNCTION
A function f A→A is said to be an
Identity Function on A denoted by IA .
f(x) = x
A
A
x
x
f:A→A
CONSTANT FUNCTION
A Function f : A→B is a constant function
if there is an element cЄB such that f(x) =c
A
1
B
f
a
b
2
3
c
d
COMPOSITE FUNCTION

Let F:A→B G:B→C be two functions
then the composite function of F and
G denoted by gof.
g
f
f : A→B
g : B→C
gof :A→C
GRAPHS OF FUNCTION
Eg-2
Eg-6
Eg-1
Eg-3
Eg-5
Eg-4
O
Line
Cutsl the
cutsgraph
the graph
onceTWICE
Ex-1
Let f, g, h be functions defined as follows f(x)=(x+2);
g(x)=3x-1; h(x)= 2x
{ho[gof](x)
={h(gof)(x)}
=h{g[f(x)]}
=h[g(x+2)]
=h[3(x+2)-1]
=h(3x+5)
=2(3x+5)
=6x+10
show that ho(gof)=(hog)of
{[hog]of}(x)
=(hog)[f(x)]
=h{g[f(x)]}
=h[g(x+2)}
=h[3(x+2)-1]
=h(3x+5)
=2(3x+5)
=6x+10
ho(go) = (hog)of
.
EXERCISES
1. Sate and define types of functions.
2. Define Inverse of a function and Inverse
function.
3. Let A={-1,1}. Let the functions f1 and f2
and f3 be from A into A defined as
follows: f1(x)=x; f2(x)=x2 ; f3(x)=x3.
4. Let f(x)=x2+2, g(x)=x2-2, for xЄR , find
fog(x), gof(x).
 http://www-
history.mcs.stand.ac.uk/history/Mathe
maticians/Cantor.html
 Micro soft Encarta.
 Telugu Academy Text
Book - 10th
class.