How does the absolute value function affect the injectivity of a function?

The absolute value function, denoted as \( |x| \), significantly affects the injectivity of a function because it is not one-to-one itself. This can be demonstrated by the following properties of the absolute value function:

1. **Definition**: The absolute value of \( x \) is defined as:
   \[
   |x| = 
   \begin{cases} 
   x & \text{if } x \geq 0 \\
   -x & \text{if } x < 0 
   \end{cases}
   \]

2. **Non-injectivity**: The absolute value function maps both a positive and its corresponding negative value to the same non-negative value. For example, \( |3| = 3 \) and \( |-3| = 3 \). This means that different inputs can produce the same output, which violates the criterion for injectivity.

When the absolute value function is combined with other functions, it retains this non-injective characteristic. Consider the following examples:

### Example 1: \( f(x) = |x| \)
- For \( x = 2 \) and \( x = -2 \), \( f(2) = |2| = 2 \) and \( f(-2) = |-2| = 2 \).
- Here, \( f(2) = f(-2) \), indicating that \( f(x) \) is not one-to-one.

### Example 2: \( f(x) = |x + 1| - 20 \)
- For \( x = 1 \) and \( x = -3 \), \( f(1) = |1 + 1| - 20 = 2 - 20 = -18 \) and \( f(-3) = |-3 + 1| - 20 = |-2| - 20 = 2 - 20 = -18 \).
- Again, \( f(1) = f(-3) \), so \( f(x) \) is not one-to-one.

### General Impact
- **Breaking Monotonicity**: Functions that might be monotonic (strictly increasing or decreasing) over certain intervals lose this property when composed with the absolute value function. This breaking of monotonicity means that horizontal lines can intersect the graph of the function more than once, violating the horizontal line test for injectivity.

### Conclusion
The absolute value function introduces symmetry about the y-axis, causing different inputs (one positive and one negative) to produce the same output. This fundamental characteristic ensures that any function incorporating the absolute value function cannot be one-to-one.

Would you like any more details or have any questions? Here are some more questions you might be interested in:

1. Can you provide examples of injective functions that do not involve the absolute value?
2. How do you graphically determine if a function is one-to-one?
3. What are the implications of a function being one-to-one in real-world applications?
4. Can you explain the difference between injective, surjective, and bijective functions?
5. How can you algebraically verify if a function is one-to-one without graphing it?

**Tip:** When working with absolute value functions, always consider their behavior over different intervals (positive and negative domains) to understand their impact on the overall function's injectivity.

Explore how the absolute value function affects the injectivity of a function. Understand why functions including the absolute value are not one-to-one, with examples and detailed explanations. Suitable for high school students studying function properties.

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