The problem involves determining the image of various sets under the map $W = \frac{1}{2}$, which is a transformation on the complex plane. We will proceed with each subquestion and compute the corresponding images under this map.

General Approach: The map $W = \frac{1}{2}$ means we are transforming the complex number $z$ into a new number, which is always half of the original value of $z$. Hence, if $z$ is mapped to $W = \frac{1}{2}$, the image of any point or set depends on how this transformation affects the geometric properties of that set.

Let me now analyze each part of the question:

(a) $|z + 1| = 1$

This describes a circle centered at $z = -1$ with radius 1. Under the map, each point on this circle is mapped to $W = \frac{1}{2}(z)$. The image will also be a circle, but with a new center and radius scaled by $\frac{1}{2}$. The new circle will have center $W = \frac{1}{2}(-1) = -\frac{1}{2}$, and radius $\frac{1}{2}(1) = \frac{1}{2}$.

(b) $|z - 2i| = 2$

This describes a circle centered at $z = 2i$ with radius 2. After applying the map, the new center is $W = \frac{1}{2}(2i) = i$, and the new radius is $\frac{1}{2}(2) = 1$. Thus, the image is a circle with center $i$ and radius 1.

(c) $x = 1$

This represents the vertical line in the complex plane where the real part of $z$ is 1. The transformation scales the real part of $z$ by $\frac{1}{2}$. Hence, the image will be the line where the real part of $W$ is $\frac{1}{2}$, i.e., the line $x = \frac{1}{2}$.

(d) $1 < x < 2$

This describes a vertical strip where the real part of $z$ lies between 1 and 2. Under the map, the real part of $z$ is halved, so the image is a strip where $\frac{1}{2} < x < 1$.

(e) $x = 2$

This is a vertical line where the real part of $z$ is 2. After applying the map, the real part of $W$ becomes $\frac{1}{2}(2) = 1$, so the image is the line $x = 1$.

(f) $|z - 1| = 1$

This describes a circle centered at $z = 1$ with radius 1. After the transformation, the center becomes $W = \frac{1}{2}(1) = \frac{1}{2}$, and the radius becomes $\frac{1}{2}(1) = \frac{1}{2}$. Hence, the image is a circle centered at $\frac{1}{2}$ with radius $\frac{1}{2}$.

Would you like more details on any of these transformations or additional explanations for the geometric reasoning?