Find the range of $f(z)=|z|$ defined on the square $0 \leq \Re z \leq 1$, $0 \leq \Im z \leq 1$.
I suspect this function maps each point in or on the square into the value of its modulus. So would the range be the line segment on the real axis connecting the origin with the point $z=\sqrt{2}$? I am seeing other answers and I do not know if I am correct.
Let $z=x+yi$. Then,
$$f(z) = |z|=\sqrt{x^2+y^2}$$
Given that $0\le x\le1$ and $0\le y\le1$,
$$0=f(0)\le f(z) \le f(1+i) = \sqrt2$$