Functions

Given,

f(x)%3D%5Cfrac%7B1%7D%7B2-%5Ccos%20%5C%2C3x%7D%2C%5C%2Cx%5C%2C%5Cin%20%5Cleft%5B%200%2C%5Cfrac%7B%5Cpi%20%7D%7B3%7D%20%5Cright%5D

For one - one Let

f(%7B%7Bx%7D_%7B1%7D%7D)%3Df(%7B%7Bx%7D_%7B2%7D%7D)

%5CRightarrow

%5Cfrac%7B1%7D%7B2-%5Ccos%20%5C%2C3%7B%7Bx%7D_%7B1%7D%7D%7D%3D%5Cfrac%7B1%7D%7B2-%5Ccos%20%5C%2C3%5C%2C%7B%7Bx%7D_%7B2%7D%7D%7D

%5CRightarrow

2-%5Ccos%20%5C%2C3%7B%7Bx%7D_%7B1%7D%7D%3D2-%5Ccos%20%5C%2C3%7B%7Bx%7D_%7B2%7D%7D

%5CRightarrow

%5Ccos%20%5C%2C3%7B%7Bx%7D_%7B1%7D%7D%3D%5Ccos%20%5C%2C3%7B%7Bx%7D_%7B2%7D%7D%5C%2C%5C%2C%5CRightarrow%20%5C%2C%5C%2C%7B%7Bx%7D_%7B1%7D%7D%3D%7B%7Bx%7D_%7B2%7D%7D

%5CRightarrow

f is one-one For onto Let

y%3Df(x)%2C%5C%2C%5C%2Cy%5C%2C%5C%2C%5Cin

codomain

%5CRightarrow

y%3D%5Cfrac%7B1%7D%7B2-%5Ccos%20%5C%2C3x%7D

%5CRightarrow

y(2-%5Ccos%20%5C%2C3x)%3D1

%5CRightarrow

2-%5Ccos%20%5C%2C3x%3D%5Cfrac%7B1%7D%7By%7D

%5CRightarrow

%5Ccos%20%5C%2C3x%3D2-%5Cfrac%7B1%7D%7By%7D

%5CRightarrow

x%3D%5Cfrac%7B1%7D%7B3%7D%5C%2C%7B%7B%5Ccos%20%7D%5E%7B-1%7D%7D%5Cleft(%202-%5Cfrac%7B1%7D%7By%7D%20%5Cright)

Here, for all

y%5C%2C%5Cin

codomain there exist

x%5C%2C%5Cin

domain. so

f(x)

is onto.
Here for all

y%5C%2CE

codomain there exist

x%5C%2C%5Cin

domain. so is on to.

f(x)

Chapter Questions
3 MCQs