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The question is about finding intercepts and asymptotes for the rational function:

\[
f(x) = \frac{3x}{x - 5}
\]

Let's break down the missing answer step by step:

1. **Y-Intercept**:
   - To find the y-intercept, set \( x = 0 \) and evaluate \( f(0) \):
     \[
     f(0) = \frac{3 \cdot 0}{0 - 5} = 0
     \]
     Therefore, the y-intercept is \( 0 \).
   
2. **X-Intercept** (already answered):
   - To find the x-intercept, set \( f(x) = 0 \):
     \[
     \frac{3x}{x - 5} = 0 \Rightarrow x = 0
     \]
     So, the x-intercept is \( x = 0 \).

3. **Vertical Asymptote** (already answered):
   - The vertical asymptote occurs where the denominator is zero:
     \[
     x - 5 = 0 \Rightarrow x = 5
     \]
     So, the vertical asymptote is \( x = 5 \).

4. **Horizontal Asymptote** (missing answer):
   - For horizontal asymptotes, compare the degrees of the numerator and denominator.
   - Both numerator and denominator have degree 1, so divide the leading coefficients:
     \[
     y = \frac{3}{1} = 3
     \]
     Thus, the horizontal asymptote is \( y = 3 \).

So, the missing answer in the box for the horizontal asymptote should be:
\[
y = 3
\]

Would you like more details or have any questions?

---

**Follow-up Questions:**
1. How would the graph of \( f(x) = \frac{3x}{x - 5} \) change if the numerator was different?
2. What does the existence of a vertical asymptote at \( x = 5 \) tell us about the function's behavior?
3. How does the horizontal asymptote affect the end behavior of \( f(x) \)?
4. What would happen to the intercepts if we changed the function to \( f(x) = \frac{3x + 1}{x - 5} \)?
5. How can we verify intercepts and asymptotes using a graphing calculator?

**Tip:** For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator.

Find the intercepts and asymptotes for the rational function f(x) = 3x / (x - 5).

Learn how to find the intercepts and asymptotes for the rational function f(x) = 3x / (x - 5). This solution includes step-by-step calculations for finding x-intercepts, y-intercepts, vertical asymptotes, and horizontal asymptotes. Suitable for high school students in Grades 10-12.

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