This exercise involves complex numbers and their geometric interpretation in the complex plane. Let's go step by step to break down the questions:

a) Show that: $Z = f(Z) = \frac{Z + i}{Z + 1}$

The goal of this part is to prove a relationship involving the complex number $Z$ and its transformation. You are given that $Z = f(Z)$, and you need to derive the expression from the provided form $Z = \frac{Z + i}{Z + 1}$.

b) Write $Z$ in exponential form.

You are asked to express the complex number $Z$ in its exponential form, which generally takes the form: $Z = r e^{i \alpha}$ where $r$ is the modulus (absolute value) of $Z$, and $\alpha$ is its argument.

c) Demonstrate that the set of invariant points is the line $y = x$.

This question likely refers to showing that a set of points remains invariant under the transformation $f(Z)$. You will need to prove that the geometric locus of points that do not change under $f(Z)$ lies on the line $y = x$ in the complex plane.

d) Prove that the set $M$ is the orthogonal projection of $M'$ on the line $y = x$.

You are expected to prove that $M$ is the orthogonal projection of another set of points $M'$ onto the line $y = x$. This involves geometric reasoning and orthogonal projections in the complex plane.

e) Verify that if $M \in M(d)$, then $(MM') \perp (OA)$.

In this part, you are required to verify that certain geometric conditions hold. Specifically, you need to show that the line joining $M$ to $M'$ is perpendicular to the line joining $O$ and $A$.

f) Determine the set of points $M$ such that $|Z| = \sqrt{2}$.

This is a geometric problem, asking you to find the set of points $M$ whose modulus $|Z|$ is equal to $\sqrt{2}$.

g) Determine the set of points $M$ such that $|Z + Z_0| = 8$.

This part asks you to find the locus of points for which the sum of $Z$ and a fixed complex number $Z_0$ has a modulus of 8.

h) Show that $Z = \frac{1}{Z}$.

Here, you need to demonstrate that $Z$ satisfies the equation $Z = \frac{1}{Z}$, which often leads to a simple equation for $Z$, possibly involving its modulus and argument.

Would you like to go into more detail on a specific part, or should I start solving one of these questions? Let me know which part you’d like to focus on!