the domain and range of a function
Every function comes with two sets that are as much a part of it as its formula. The domain is the collection of inputs the function is allowed to accept; the range (or image) is the collection of outputs it actually produces as z runs over the whole domain. Asking 'what is f?' without saying what its domain is is like quoting a price without a currency — incomplete.
For a complex function w = f(z), the domain is a set of points in the z-plane and the range is a set of points in the w-plane. You find the range by imagining z visiting every point of the domain and collecting all the resulting w's. For example, f(z) = 1/z has domain 'all z except 0' (you cannot divide by zero), and its range turns out to be 'all w except 0' as well. As another, f(z) = z^2 with domain the whole plane has range the whole plane, because every complex number has a square root.
Two cautions worth keeping. First, the range can be much smaller than the whole w-plane — the constant function f(z) = 3 has range the single point 3. Second, in complex analysis the word 'domain' has a second, technical meaning: an open connected set (a region you can move around inside without leaving). Context tells you which is meant; here we mean the set of allowed inputs, but the two ideas often coincide because we usually study functions on open connected sets.
f(z) = sqrt(|z|) is fine for every z, so its domain is the whole plane; but the output is a non-negative real number, so its range is only the non-negative real axis — a tiny sliver of the w-plane.
The range answers 'which outputs really occur?' and is often far smaller than the entire target plane.
Do not confuse range (outputs that actually occur) with codomain (the plane the outputs are allowed to live in). Saying 'f maps into the plane' names the codomain; the range may fill only part of it.