If $f: R \rightarrow C$ is defined by $f(x)=e^{2 i x}$ for $x \in R$, then $f$ is (where $C$ denotes the set…
If $f: R \rightarrow C$ is defined by $f(x)=e^{2 i x}$ for $x \in R$, then $f$ is (where $C$ denotes the set of all complex numbers)
one-one
onto
one-one and onto
neither one-one nor onto
Solution
Given that, $f(x)=e^{2 i x}$ and $f: R \rightarrow C$. Function $f(x)$ is not one-one, because after some values of $x(i e, \pi)$ it will give the same values.
Also, $f(x)$ is not onto, because it has minimum and maximum values $-1-i$ and $1+i$ respectively.
Hence, option (d) is correct.