The function \( f(x) = -|x| \) can be analyzed as follows:

### Definition:
- The absolute value function \( |x| \) is defined as:
  \[
  |x| = \begin{cases}
  x & \text{if } x \geq 0 \\
  -x & \text{if } x < 0
  \end{cases}
  \]
- Applying the negative sign to \( |x| \) means:
  \[
  f(x) = \begin{cases}
  -x & \text{if } x \geq 0 \\
  x & \text{if } x < 0
  \end{cases}
  \]

### Characteristics:
- **Piecewise Function**:
  \[
  f(x) = \begin{cases}
  -x & \text{if } x \geq 0 \\
  x & \text{if } x < 0
  \end{cases}
  \]
- **Graph**:
  - The graph of \( f(x) = -|x| \) is an upside-down V-shape. The vertex is at the origin \((0, 0)\).
  - For \( x \geq 0 \), the graph decreases linearly with a slope of \(-1\).
  - For \( x < 0 \), the graph increases linearly with a slope of \(1\).

### Properties:
- **Domain**: The function is defined for all real numbers, so the domain is \( (-\infty, \infty) \).
- **Range**: The range of \( f(x) = -|x| \) is \( (-\infty, 0] \) since the function takes values from \(0\) (inclusive) downwards.
- **Continuity**: The function is continuous for all \( x \).
- **Differentiability**: The function is differentiable everywhere except at \( x = 0 \) where there is a sharp corner.

### Conclusion:
- \( f(x) = -|x| \) describes a simple, continuous piecewise linear function with a distinctive V-shaped graph.

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Here are some related questions:
1. How do you graph the function \( f(x) = |x| \)?
2. What is the derivative of \( f(x) = -|x| \) for \( x \neq 0 \)?
3. How does the graph of \( f(x) = -|x| + c \) differ from \( f(x) = -|x| \)?
4. What is the integral of \( f(x) = -|x| \) over a specific interval?
5. How does the function \( g(x) = |x| - 1 \) compare to \( f(x) = -|x| \)?
6. Can you solve \( f(x) = -|x| \) for \( x \) when \( f(x) = -3 \)?
7. What are the applications of the absolute value function in real life?
8. How does the function change if we have \( f(x) = -|2x| \)?

**Tip:** When dealing with absolute value functions, remember that they create piecewise linear segments on a graph.

Explore the function f(x) = -|x| with detailed analysis and graphical representation. Learn about its domain, range, continuity, and differentiability. Suitable for high school students and anyone interested in mathematical functions.

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