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

1 Answer
George C.
Apr 21, 2017

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.

Related questions
Impact of this question
882 views around the world
You can reuse this answer
Creative Commons License