Question

We have `f:RR->CC`*,`f(x)=cos(2xpi)+isin(2xpi)`.How to demonstrate that `f` is not surjective?

Answer 1

`f` maps all real numbers into the unit circle, so is not surjective onto `CC`.

Explanation:

Given:

`f(x) = cos(2xpi)+isin(2xpi)`

Note that:

`abs(f(x)) = sqrt(cos^2(2xpi)+sin^2(2xpi)) = sqrt(1) = 1`

So `f` maps any real number into the unit circle. So any number that is not on the unit circle is not in the range of `f`. So `f` is not a surjection onto `CC`.