5.6.12. Let f : [0, 2] → [5, 6] be a function.
132 ⃝c 2017 Shay Fuchs. All rights reserved.
CHAPTER 5. BIJECTIONS AND CARDINALITY 5.6. EXERCISES FOR CHAPTER 5
(a) If the restrictions of f to [0, 1] and to [1, 2] are injective functions, must f be an injection? Explain.
(b) If the restrictions of f to [0, 1] and to [1, 2] are surjective functions, must f be a surjection? Explain.

Let us address the two parts of this problem systematically. We analyze the properties of injectivity and surjectivity for the function \( f: [0, 2] \to [5, 6] \), given the conditions.

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### (a) **If the restrictions of \( f \) to \([0, 1]\) and \([1, 2]\) are injective, must \( f \) be an injection?**

An **injective function** (or one-to-one function) ensures that every element in the domain maps to a unique element in the codomain. In mathematical terms:
\[
f(x_1) = f(x_2) \implies x_1 = x_2.
\]

- **Given Information:**
    - \( f \) is injective on \([0, 1]\), meaning no two points in \([0, 1]\) map to the same value.
    - \( f \) is injective on \([1, 2]\), meaning no two points in \([1, 2]\) map to the same value.

- **Analysis:**
    - Injectivity on each subinterval does not guarantee that the same output value cannot occur in both \([0, 1]\) and \([1, 2]\). That is, there might exist \( x_1 \in [0, 1] \) and \( x_2 \in [1, 2] \) such that \( f(x_1) = f(x_2) \). 
    - If such a case occurs, \( f \) is no longer injective on the entire domain \([0, 2]\), even though it is injective on each subinterval.

- **Counterexample:**
    Define \( f(x) \) as:
    \[
    f(x) = \begin{cases}
    5.5 - 0.5x, & \text{if } x \in [0, 1], \\
    5.5 + 0.5(x - 1), & \text{if } x \in [1, 2].
    \end{cases}
    \]
    Here, \( f(x) \) is injective on \([0, 1]\) and \([1, 2]\), but \( f(0) = f(2) = 5.5 \), which means \( f \) is not injective on \([0, 2]\).

- **Conclusion:**
    The function \( f \) is **not necessarily injective**.

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### (b) **If the restrictions of \( f \) to \([0, 1]\) and \([1, 2]\) are surjective, must \( f \) be a surjection?**

A **surjective function** (or onto function) ensures that every element in the codomain has at least one corresponding element in the domain. In mathematical terms:
\[
\forall y \in [5, 6], \exists x \in [0, 2] \text{ such that } f(x) = y.
\]

- **Given Information:**
    - \( f \) is surjective on \([0, 1]\), meaning every value in \([5, 6]\) can be attained using \( x \in [0, 1] \).
    - \( f \) is surjective on \([1, 2]\), meaning every value in \([5, 6]\) can be attained using \( x \in [1, 2] \).

- **Analysis:**
    - Surjectivity on both subintervals ensures that every \( y \in [5, 6] \) has at least one preimage in \([0, 1]\) and at least one preimage in \([1, 2]\). 
    - Thus, \( f \) is surjective on the entire domain \([0, 2]\), because \([0, 1] \cup [1, 2] = [0, 2]\).

- **Counterexample Possibility:**
    There is no way to construct a counterexample where \( f \) fails to be surjective, because the union of the surjective mappings on \([0, 1]\) and \([1, 2]\) covers the entire codomain \([5, 6]\).

- **Conclusion:**
    The function \( f \) **must be surjective**.

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### Final Summary:
(a) \( f \) is not necessarily injective.  
(b) \( f \) must be surjective.

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Do you want a detailed visualization or examples?  
Here are 5 related questions to explore:

1. What conditions ensure injectivity on a domain partitioned into subintervals?
2. Can you define a function that is bijective when its restrictions are bijections?
3. How does the codomain affect the injectivity or surjectivity of a function?
4. What are practical applications of injective and surjective functions in real-world problems?
5. How does bijection relate to cardinality?

**Tip:** Always analyze how subinterval restrictions influence the global behavior of a function when combining properties like injectivity and surjectivity.

Let f : [0, 2] → [5, 6] be a function. (a) If the restrictions of f to [0, 1] and to [1, 2] are injective functions, must f be an injection? Explain. (b) If the restrictions of f to [0, 1] and to [1, 2] are surjective functions, must f be a surjection? Explain.

Explore whether a function defined on [0, 2] → [5, 6] is injective or surjective based on its restrictions to subintervals. Learn the concepts of injectivity, surjectivity, and their applications in function analysis.

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