by sampath ix class 1 2 3 a + - f g gof georg ferdinand ludwig philipp cantor born: 3 march 1845 in...
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BYBY
SampathSampath
Ix classIx class
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Georg FerdinandLudwig Philipp CantorBorn: 3 March 1845in St Petersburg,Russia Died: 6 Jan 1918 in Halle, Germany
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GEORG CANTOR born in St petersburg, RUSSIA
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TYPES OF FUNCTIONTYPES OF FUNCTION
One to One FunctionOne to One Function On to FunctionOn to Function One to One On to One to One On to Inverse of a FunctionInverse of a Function Equal FunctionEqual Function Identity FunctionIdentity Function Constant FunctionConstant Function Composite FunctionComposite Function
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O N E T O 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.
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On to FunctionOn to Function
f : A B is said to be an On to Function. If f(A) is the image of A 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 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.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 onef: A B is on to for every x € B there exist at least one
x € A such that f(x) = y f(A) = B.x € A such that f(x) = y f(A) = B.
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One to one on toOne to one on to
A function f : A B is said to be a A function f : A B is said to be a bijection if it is both one to one and on to.bijection if it is both one to one and on to.
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INVERSE FUNCTIONINVERSE FUNCTION If f is a function then the set of If f is a function then the set of
ordered pairs obtained by ordered pairs obtained by interchanging the first and second interchanging the first and second coordinates of each order fair in F s coordinates of each order fair in F s called inverse of F. it denoted by Fcalled inverse of F. it denoted by F-1-1
f = { (0,0), (1,1), (2,4), (2,9)……..} f = { (0,0), (1,1), (2,4), (2,9)……..} f f -1-1 = { (0,0), (1,1), (4,2), (9,2) = { (0,0), (1,1), (4,2), (9,2)……..}……..}
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IDENTITY FUNCTION
A function f A→A is said to be an Identity Function on A denoted by IA .
f(x) = x
x x
f : A → A
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CONSTANT FUNCTIONCONSTANT FUNCTION
A Function f : A→B is a constant A Function f : A→B is a constant function if there is an element cfunction if there is an element cЄЄB B such that f(x) =csuch that f(x) =c
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COMPOSITE FUNCTIONCOMPOSITE FUNCTION
Let F:A→B G:B→C be two functions Let F:A→B G:B→C be two functions then the composite function of F and then the composite function of F and G denoted by gof.G denoted by gof.
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f : A→B g : B→C
gof :A→C
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GRAPHS OF FUNCTION
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Eg-1
Cuts the graph once
Eg-2 Eg-3Eg-4Eg-5Eg-6
Line l cuts the graph TWICE
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Let f, g, h be functions defined as follows Let f, g, h be functions defined as follows f(x)=(x+2); g(x)=3x-1; h(x)= 2x show that f(x)=(x+2); g(x)=3x-1; h(x)= 2x show that
ho(gof)=(hog)ofho(gof)=(hog)of . . {ho[gof](x) {[hog]of}(x){ho[gof](x) {[hog]of}(x)={h(gof)(x)} =(hog)[f(x)]={h(gof)(x)} =(hog)[f(x)]=h{g[f(x)]} =h{g[f(x)]}=h{g[f(x)]} =h{g[f(x)]}=h[g(x+2)] =h[g(x+2)}=h[g(x+2)] =h[g(x+2)}=h[3(x+2)-1] =h[3(x+2)-1] =h[3(x+2)-1] =h[3(x+2)-1] =h(3x+5) =h(3x+5)=h(3x+5) =h(3x+5)=2(3x+5) =2(3x+5)=2(3x+5) =2(3x+5)=6x+10 =6x+10=6x+10 =6x+10 ho(go) = (hog)ofho(go) = (hog)of
Ex-1
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EXERCISESEXERCISES
1.1. Sate and define types of functions.Sate and define types of functions.
2.2. Define Inverse of a function and Inverse Define Inverse of a function and Inverse function.function.
3.3. Let A={-1,1}. Let the functions fLet A={-1,1}. Let the functions f11 and f and f22
and f3 be from A into A defined as and f3 be from A into A defined as follows: ffollows: f11(x)=x; f(x)=x; f22(x)=x(x)=x22 ; f ; f33(x)=x(x)=x33..
4.4. Let f(x)=x2+2, g(x)=x2-2, for xLet f(x)=x2+2, g(x)=x2-2, for xЄЄR , find R , find fog(x), gof(x).fog(x), gof(x).
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http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Cantor.html
Micro soft Encarta. Telugu Academy Text
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