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Q.

If  f:RCisdefinedbyf(x)=e2ixxR,thenfis(whereCdenotesthesetofallcomplexnumbers)

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a

Onto

b

Neither one – one nor onto

c

bijection

d

One – one

answer is D.

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Detailed Solution

f(x)=e2ix=cos2x+isin2xweknowthatcosx  and  sinx  areperiodicfunctionsin2x  and  cos2x  period  =2ππ=π  f(π+x)=cos2(π+x)+isin2(π+x)=cos2x+isin2x=f(x)  Periodicfunctionsarenotoneone|f(x)|=|e2ix|=|cos2x+isin2x|=1

But in complex plane, magnitude of a complex number can vary from 0 up to infinity.

For any value of x magnitude of f(x)

Can not be greater than or less than ‘1’

f(x) does not cover the entire complex number

f is not onto.

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