Give an example of when a non-injective function would fail

Let's consider a specific example to illustrate how a non-injective function can cause the equation \( f(A \cap B) = f(A) \cap f(B) \) to fail.

### Example:

Let \( f: \mathbb{R} \to \mathbb{R} \) be the function defined by \( f(x) = x^2 \). This function is not injective because different inputs can produce the same output. For example, \( f(2) = 4 \) and \( f(-2) = 4 \).

Now, let's define two sets \( A \) and \( B \) as follows:
- \( A = \{-2, 1\} \)
- \( B = \{2, 1\} \)

#### Step 1: Find \( A \cap B \)

The intersection of \( A \) and \( B \) is:
\[
A \cap B = \{1\}
\]

#### Step 2: Find \( f(A \cap B) \)

Now, apply the function \( f(x) = x^2 \) to the intersection:
\[
f(A \cap B) = f(\{1\}) = \{1^2\} = \{1\}
\]

#### Step 3: Find \( f(A) \) and \( f(B) \)

Next, find the images of sets \( A \) and \( B \) under \( f \):
\[
f(A) = f(\{-2, 1\}) = \{(-2)^2, 1^2\} = \{4, 1\}
\]
\[
f(B) = f(\{2, 1\}) = \{2^2, 1^2\} = \{4, 1\}
\]

#### Step 4: Find \( f(A) \cap f(B) \)

The intersection of \( f(A) \) and \( f(B) \) is:
\[
f(A) \cap f(B) = \{4, 1\} \cap \{4, 1\} = \{4, 1\}
\]

#### Conclusion: 

Now, compare \( f(A \cap B) \) with \( f(A) \cap f(B) \):
\[
f(A \cap B) = \{1\}
\]
\[
f(A) \cap f(B) = \{4, 1\}
\]

Clearly, \( f(A \cap B) \neq f(A) \cap f(B) \), showing that the equation fails when \( f \) is not injective.

Would you like to explore more about this, or do you have any other questions?

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### Related Questions:
1. How do injective functions affect set operations like union and difference?
2. What are some other examples of non-injective functions?
3. How can you determine if a function is injective from its graph?
4. Can a function be surjective but not injective? Provide an example.
5. How does bijectivity ensure the equality of \( f(A \cap B) = f(A) \cap f(B) \)?

### Tip:
When dealing with functions and set operations, constructing specific examples with small, manageable sets can help clarify how function properties like injectivity affect the outcome.

Explore an example illustrating how non-injective functions affect set operations. Learn about functions, injectivity, and set theory concepts with detailed steps and comparisons. Suitable for advanced high school students.

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