Is $f(z)=e^z$ bounded on the imaginary axis?
My attempt: I know $f(z) = e^z$ is nonconstant entire function because its derivative
$\frac{\partial e^z}{\partial z} = e^z$ ...
When I take $z = x + iy$ then I got
$f(z) = e^z = e^{x+iy} = e^x \cdot e^{iy} = e^x \cdot (\cos(x) + i \sin(y)) =e^x \cos(x) + i (e^x) \sin(x)$
I know that $e^x$ is unbounded so my answer is that it will unbounded on the imaginary axis.
Is my answer correct or not, please verify my mistakes