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.