Image (of a set)

The set of all output values $f(z)$ as $z$ ranges over a given set.

The image of a set $S$ under a function $f$ , written $f(S)$ , is the set of every output value $f(z)$ obtained as $z$ ranges over $S$ . Since a complex function turns points into points, its image is a new region of the plane -- picturing how $f$ reshapes a disk, a line, or a half-plane into its image is one of the main ways this course builds intuition for what a function 'does.'

Examples

Under $f(z)=z^2$ , the image of the positive real axis is again the positive real axis.
Under $f(z)=iz$ (a $90°$ rotation), the image of the right half-plane $\operatorname{Re}(z)>0$ is the upper half-plane $\operatorname{Im}(z)>0$ .