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The exercise shown in the image (Exercise 3.2.7) asks to demonstrate that the following functions are **one-to-one** (injective), using the definition of one-to-one and Proposition 3.2.4. 

The functions are as follows:

1. \( f: [0, \infty) \to \mathbb{R}, \, f(x) = x^2 \)
2. \( f: \mathbb{Z} \to \mathbb{Z}, \, f(k) = 3k + 7 \)
3. \( f: \mathbb{N} \to \mathbb{N}, \, f(n) = \begin{cases} n+2 & \text{if } n \text{ is odd} \\ 2n & \text{if } n \text{ is even} \end{cases} \)

To prove injectivity (one-to-one), we need to show that for any two inputs \( x_1 \) and \( x_2 \), if \( f(x_1) = f(x_2) \), then it must follow that \( x_1 = x_2 \).

### Approach:

1. **For \( f(x) = x^2 \), \( x \in [0, \infty) \):**
   \[
   f(x_1) = f(x_2) \implies x_1^2 = x_2^2
   \]
   Since \( x_1 \) and \( x_2 \) are non-negative (due to the domain being \( [0, \infty) \)), this implies:
   \[
   x_1 = x_2
   \]
   Therefore, \( f(x) = x^2 \) is injective on \( [0, \infty) \).

2. **For \( f(k) = 3k + 7 \), \( k \in \mathbb{Z} \):**
   \[
   f(k_1) = f(k_2) \implies 3k_1 + 7 = 3k_2 + 7
   \]
   Simplifying:
   \[
   3k_1 = 3k_2 \implies k_1 = k_2
   \]
   Therefore, \( f(k) = 3k + 7 \) is injective on \( \mathbb{Z} \).

3. **For the piecewise function:**
   - For odd \( n \), \( f(n) = n + 2 \):
     \[
     f(n_1) = f(n_2) \implies n_1 + 2 = n_2 + 2 \implies n_1 = n_2
     \]
     Hence, the function is injective for odd \( n \).
   
   - For even \( n \), \( f(n) = 2n \):
     \[
     f(n_1) = f(n_2) \implies 2n_1 = 2n_2 \implies n_1 = n_2
     \]
     Therefore, the function is injective for even \( n \).

Since both parts of the piecewise function are injective, the whole function is injective.

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Would you like more detailed explanations on any part? Here are some related questions to expand your understanding:

1. What is the definition of a one-to-one (injective) function?
2. How does the injectivity of \( f(x) = x^2 \) change if the domain is the entire real line?
3. Why is the function \( f(k) = 3k + 7 \) injective for all integers?
4. Can a function be injective but not surjective? Provide examples.
5. What are the implications of injectivity in the context of inverse functions?

**Tip:** To check if a function is injective, always set \( f(x_1) = f(x_2) \) and try to show that this implies \( x_1 = x_2 \).

Show that the following functions are one-to-one, using the definition and Proposition 3.2.4: (i) f: [0, ∞) → ℝ, f(x) = x², (ii) f: ℤ → ℤ, f(k) = 3k + 7, (iii) f: ℕ → ℕ, f(n) = n + 2 if n is odd and f(n) = 2n if n is even.

Learn how to prove the injectivity (one-to-one property) of functions, including quadratic, linear, and piecewise functions. Using Proposition 3.2.4, this problem covers the steps to demonstrate injectivity for different types of functions. Suitable for university-level introductory math students.

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