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?

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

Hi Menka,

Here's the answer in the scanned copy:

Hope that helps.

All the best.

Regards,

Ashwin (IIT Madras).

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