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Let's analyze and provide solutions to each exercise outlined in the image.

### Exercise 1
You are asked to study the injectivity, surjectivity, and bijectivity of the following functions:

#### 1.1 \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = x^2 + x + 2 \).
- **Injectivity**: To check if \( f \) is injective, we need to see if \( f(x_1) = f(x_2) \) implies \( x_1 = x_2 \).
- **Surjectivity**: To check if \( f \) is surjective, we examine whether every real number \( y \in \mathbb{R} \) has a corresponding \( x \) such that \( f(x) = y \).
- **Bijectivity**: \( f \) is bijective if it is both injective and surjective.

#### 1.2 \( g : [1, +\infty[ \to [2, +\infty[ \) defined by \( g(x) = \frac{2x}{x-1} \).
- Similar steps: Check injectivity by examining if \( g(x_1) = g(x_2) \) leads to \( x_1 = x_2 \), and check surjectivity by finding the range of \( g(x) \).

#### 1.3 \( h : [-2, -1] \to [0, 1] \) defined by \( h(x) = \frac{x^2}{x+1} \).
- This function is limited to a specific interval, so it’s necessary to check the behavior within that interval for injectivity and surjectivity.

#### 1.4 \( \varphi : \mathbb{R}^2 \to \mathbb{R} \) defined by \( \varphi(x, y) = x - y^3 \).
- For injectivity, we check if \( \varphi(x_1, y_1) = \varphi(x_2, y_2) \) implies \( (x_1, y_1) = (x_2, y_2) \).
- Surjectivity can be analyzed by determining if all values in \( \mathbb{R} \) are attainable by the function.

### Exercise 2
This exercise involves several parts:

#### 2.1 Function \( f : \mathbb{N} \to \mathbb{Q} \) with a given formula.
   - **Determine** \( f(1), f(2), f(3) \).
   - **Injectivity**: Show that \( f(p) = f(q) \Rightarrow p = q \).
   - **Conclusion**: Deduce if \( f \) is injective.

#### 2.2 Function \( g : \mathbb{R} \to \mathbb{R} \) defined as \( g(x) = \sqrt{x} + \sqrt{x+1} \).
   - **Injectivity**: Check if \( g(x_1) = g(x_2) \Rightarrow x_1 = x_2 \).
   - **Equation Solving**: Solve \( g(x) = 2 \) to explore surjectivity.

#### 2.3 Function \( h : ]0, +\infty[ \to [1, +\infty[ \) defined as \( h(x) = \frac{1}{x} + x \).
   - **Bijectivity**: Determine if \( h \) is bijective and find the inverse \( h^{-1} \) if it exists.

### Exercise 4
Function \( f : \mathbb{Z} \times \mathbb{N}^* \to \mathbb{Q} \) with a specific mapping \( (p, q) \mapsto p + \frac{1}{q} \).

#### 4.1 Show statements involving values of \( f(p, q) \) and surjectivity.
   - **Part (a)**: Given certain conditions, deduce values of \( p \) and \( q \) based on the functional output.
   - **Part (b)**: Conclude on the surjectivity of \( f \).

#### 4.2 Check the injectivity of \( f \) on a specific subset.
   - **Part (a)**: Given \( f(p, q) = f(p', q') \), deduce if \( p = p' \).
   - **Part (b)**: Prove injectivity.

Would you like a detailed solution for each subpart, or do you want help with a specific section?

### Further Questions for Practice:
1. What conditions ensure that a function is both injective and surjective?
2. How can you use derivatives to determine if a function is injective or strictly monotonic?
3. What is the general method for finding the inverse of a bijective function?
4. How do changes in the domain and codomain of a function affect its injectivity and surjectivity?
5. What are practical applications of injective and surjective functions in real-world scenarios?

### Tip:
When analyzing functions for injectivity and surjectivity, sketching the graph can often provide valuable insights into their properties.

Study the injectivity, surjectivity, and bijectivity of the following functions. Solve exercises on injective, surjective, and bijective properties of functions, and prove injectivity or surjectivity in specific cases.

Explore detailed exercises on injective, surjective, and bijective functions. Study the properties of functions, such as f(x) = x^2 + x + 2 and g(x) = 2x / (x - 1), to determine injectivity, surjectivity, and bijectivity. Suitable for university-level mathematics students.

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