Given the function \( f(x) = \frac{4}{x} \), let's analyze its key characteristics:

1. **Domain**: The function \( f(x) = \frac{4}{x} \) is undefined when \( x = 0 \) because division by zero is undefined. Therefore, the domain is all real numbers except zero, i.e., \( x \in \mathbb{R} \setminus \{0\} \).

2. **Range**: The function can take any real value except zero, as there is no value of \( x \) that will make \( \frac{4}{x} = 0 \). Hence, the range is also \( y \in \mathbb{R} \setminus \{0\} \).

3. **Asymptotes**:
   - **Vertical asymptote**: At \( x = 0 \), the function approaches infinity, so there is a vertical asymptote at \( x = 0 \).
   - **Horizontal asymptote**: As \( |x| \) becomes very large, \( f(x) \) approaches 0, meaning there is a horizontal asymptote at \( y = 0 \).

4. **Behavior**:
   - When \( x > 0 \), \( f(x) \) is positive and decreases as \( x \) increases.
   - When \( x < 0 \), \( f(x) \) is negative and increases (in absolute value) as \( x \) decreases.

5. **Symmetry**: The function is odd, meaning \( f(-x) = -f(x) \). This indicates symmetry about the origin.

Would you like to delve into any of these characteristics in more detail?

Here are five related questions:

1. What are the limits of \( f(x) = \frac{4}{x} \) as \( x \) approaches 0 from the positive and negative sides?
2. How would you find the derivative of \( f(x) = \frac{4}{x} \), and what does it represent?
3. What are the critical points of \( f(x) = \frac{4}{x} \), and what do they signify?
4. Can you graph the function \( f(x) = \frac{4}{x} \) and identify its key features?
5. How would you integrate \( f(x) = \frac{4}{x} \), and what does the integral represent?

**Tip**: Always check the domain of a function first, as it tells you where the function is defined and can be evaluated.

Explore the function f(x) = 4/x with detailed analysis of its domain, range, asymptotes, and symmetry. Learn about its behavior, limits, and graphical representation. Suitable for advanced high school students and beyond.

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