Let's go through Exercises 3 and 4 in more detail based on the image you provided.

Exercise 3

Let:

• $f(x) = \sqrt{x}$
• $g(x) = \frac{x+2}{x+3}$

Tasks:

1. Determine the domain $D_f$ and $D_g$:

   • For $f(x) = \sqrt{x}$, $x$ must be non-negative for the square root to be defined. Thus, $D_f = [0, +\infty[$.
   • For $g(x) = \frac{x+2}{x+3}$, $x \neq -3$ to avoid division by zero. Therefore, $D_g = \mathbb{R} \setminus \{-3\}$.

2. Determine the sign of $f$ and $g$:

   • Since $f(x) = \sqrt{x}$, $f(x) \geq 0$ for all $x \in D_f$.
   • For $g(x) = \frac{x+2}{x+3}$, analyze the numerator and denominator to determine intervals where $g(x)$ is positive or negative.

3. Graph $f$ and $g$ in $\mathbb{R}$ (from $0$ to $+\infty$) and deduce $f \circ g$:

   • Plot the functions over the specified interval and examine the composition $f(g(x))$.

4. Study the function $h$ defined on $[0, +\infty[$ by: $h(x) = \frac{\sqrt{x} + 2}{\sqrt{x} + 3}$

   • a) Show that $\forall x \in [0, +\infty[$, $0 < h(x) < 1$:
     This can be shown by analyzing the range of $h(x)$ for values of $x \geq 0$.

   • b) Verify that $\forall x \in [0, +\infty[$, $h(x) = (g \circ f)(x)$:
     This requires substitution to show that $h(x)$ matches the composition $g(f(x))$.

   • c) Determine the variation of $h$ on $[0, +\infty[$:
     This entails finding the derivative of $h(x)$ to study its monotonicity.

Exercise 4

1. Solve for $x \in \mathbb{R}^+$:

   • Show that $\sqrt{1 + \sqrt{x}} = 1 + \sqrt{x}$ implies $x = 0$:
     Simplify and analyze each side of the equation to show that $x$ must be zero.

2. Determine the logical value of the following proposition:
   Given $P: (\forall x \in \mathbb{R}, x^2 = 1) \Rightarrow \frac{1}{x} \in \mathbb{R}$, determine if $P$ is true or false. Justify.

Would you like to go through the solution steps for each part in detail?

Follow-Up Questions

1. How do we analyze the domain of complex functions involving square roots and fractions?
2. What are some general steps to determine the positivity or negativity of a function?
3. How does function composition affect the domain and range of the result?
4. What are the typical steps for proving inequalities involving functions?
5. What does the use of logical propositions (like in Exercise 4) tell us about mathematical reasoning?

Tip

When working with compositions of functions, always check the inner function's output range to ensure it aligns with the domain of the outer function.