The function \( f(a) = \frac{4}{a} \) is a rational function where the output \( f(a) \) depends on the value of the variable \( a \).

### Key Characteristics of \( f(a) = \frac{4}{a} \):

1. **Domain**: The domain of the function consists of all real numbers except \( a = 0 \) because division by zero is undefined. So, the domain is \( a \in \mathbb{R} \setminus \{0\} \).

2. **Range**: The range of the function is all real numbers except zero, since \( f(a) = \frac{4}{a} \) can take any real value except for 0. Therefore, the range is \( f(a) \in \mathbb{R} \setminus \{0\} \).

3. **Asymptotes**:
   - **Vertical Asymptote**: The function has a vertical asymptote at \( a = 0 \), because as \( a \) approaches 0 from either direction, \( f(a) \) grows without bound (either positively or negatively).
   - **Horizontal Asymptote**: The horizontal asymptote is \( y = 0 \). As \( a \) becomes very large in either the positive or negative direction, \( f(a) = \frac{4}{a} \) approaches 0.

4. **Behavior**:
   - For \( a > 0 \), \( f(a) = \frac{4}{a} \) is positive and decreases as \( a \) increases.
   - For \( a < 0 \), \( f(a) = \frac{4}{a} \) is negative and increases as \( a \) becomes more negative.

5. **Intercepts**:
   - **x-intercept**: There is no x-intercept because \( f(a) \) never equals zero.
   - **y-intercept**: There is no y-intercept because the function is not defined at \( a = 0 \).

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. How do you find the inverse of the function \( f(a) = \frac{4}{a} \)?
2. What is the derivative of \( f(a) = \frac{4}{a} \)?
3. How does the function \( f(a) = \frac{4}{a} \) compare to the function \( g(a) = \frac{1}{a} \)?
4. What transformations would change the graph of \( f(a) = \frac{4}{a} \)?
5. What are some practical applications of functions like \( f(a) = \frac{4}{a} \)?

**Tip**: When analyzing rational functions, always check for asymptotes and intercepts to understand their behavior fully.

Explore the rational function f(a) = 4/a, its domain, range, and asymptotes. Learn why division by zero is undefined, and how the function behaves for positive and negative values of a. Suitable for high school students studying algebra and calculus.

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