The function you provided is \( f(x) = \frac{1}{5x} \). This represents a rational function, where the variable \( x \) is in the denominator.

### Key characteristics of this function:
1. **Domain**: The domain consists of all real numbers except \( x = 0 \) because division by zero is undefined. Therefore, the domain is \( (-\infty, 0) \cup (0, \infty) \).
   
2. **Range**: The range consists of all real numbers except 0 because the function will never output 0 for any input \( x \). So, the range is also \( (-\infty, 0) \cup (0, \infty) \).

3. **Vertical asymptote**: Since \( f(x) \) becomes undefined at \( x = 0 \), there is a vertical asymptote at \( x = 0 \).

4. **Horizontal asymptote**: As \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to 0 \). So, the horizontal asymptote is \( y = 0 \).

5. **Behavior**: The function is decreasing for both positive and negative values of \( x \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the slope of the tangent line to the graph of \( f(x) \) at a specific point?
2. How do you find the inverse of \( f(x) = \frac{1}{5x} \)?
3. What happens to \( f(x) \) as \( x \to \infty \) or \( x \to -\infty \)?
4. How do you graph this function step by step?
5. Can we determine the intervals where \( f(x) \) is increasing or decreasing?

**Tip**: Rational functions often have vertical asymptotes where the denominator equals zero, and horizontal asymptotes that reflect long-term behavior.

Explore the rational function f(x) = 1/(5x), including its domain, range, vertical and horizontal asymptotes, and graph behavior. Suitable for high school students, this detailed analysis covers key concepts in rational functions and asymptotes. Great for Grades 9-12.

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