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Is the function f:R→R defined by f(x)=cos(2x+1) invertible?Give reasons.Can you modify domain and codomain of f such that f becomes invertible?

Is the function f:R→R defined by f(x)=cos(2x+1) invertible?Give reasons.Can you modify domain and codomain of f such that f becomes invertible?

Grade:12th Pass

3 Answers

jitender lakhanpal
62 Points
10 years ago

Dear Menka,

no the function is not invertible as the function f(x) is not BIJECTIVE as it is not one to one

as there are many values of x for which there exist values of -1 to 1 the range of the function

and the function is also not onto as the codomain is all real number R but the range of f(x) is -1 to 1

and for onto the codomain must be equal to range .

as we know the necessary and sufficient condition for the function to be invertible should be 

that the function must be bijective .

to make function invertible the codomain must be equal to [-1 , 1]

and domain must be 

0 <= 2x + 1 <= pie

now by this relation we get

-1/2 <= x <= (pie - 1 )/2

so the domain must be equal to -1/2 <= x <= (pie - 1 )/2

and codomain is  [-1 , 1]. to make function invertible

 

 

 

 

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jitender

Ashwin Muralidharan IIT Madras
290 Points
10 years ago

Hi Menka,

 

Here's the answer in the scanned copy:

Hope that helps.

 

All the best.

Regards,

Ashwin (IIT Madras).

jitender lakhanpal
62 Points
10 years ago

hi ashwin 

i did not get the first part of the solution was it asked 

and i think i have done the problem in simpler way

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