To analyze the features of the function \( f(x) = \frac{-3x + 15}{5x - 25} \), we will examine the following aspects:

1. **Domain**
2. **Holes**
3. **Vertical Asymptotes**
4. **Horizontal Asymptotes**
5. **Intercepts**

### 1. Domain
The domain of \( f(x) \) includes all real numbers except where the denominator is zero. To find this, set the denominator equal to zero and solve for \( x \):

\[
5x - 25 = 0
\]

\[
x = 5
\]

Thus, the domain is all real numbers except \( x = 5 \):

\[
\text{Domain: } (-\infty, 5) \cup (5, \infty)
\]

### 2. Holes
As previously discussed, the function can be factored and simplified:

\[
f(x) = \frac{-3(x - 5)}{5(x - 5)} = \frac{-3}{5} \quad \text{(except at \( x = 5 \))}
\]

Since the factor \( (x - 5) \) cancels out, there is a **hole** at \( x = 5 \). The y-coordinate of the hole can be found by plugging \( x = 5 \) into the simplified function:

\[
f(5) = \frac{-3}{5} = -0.6
\]

So, there is a hole at \( (5, -0.6) \).

### 3. Vertical Asymptotes
Vertical asymptotes occur where the function is undefined and the factor does not cancel out. Since the factor \( (x - 5) \) cancels out, there are **no vertical asymptotes**.

### 4. Horizontal Asymptotes
To determine the horizontal asymptote, compare the degrees of the numerator and the denominator:

- The degree of the numerator \( (-3x + 15) \) is 1.
- The degree of the denominator \( (5x - 25) \) is 1.

When the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients:

\[
\text{Horizontal Asymptote: } y = \frac{-3}{5} = -0.6
\]

### 5. Intercepts
- **x-intercept**: Set \( f(x) = 0 \) and solve for \( x \). Since \( f(x) \) is a fraction, set the numerator equal to zero:

  \[
  -3x + 15 = 0 \implies x = 5
  \]

  However, since there is a hole at \( x = 5 \), there is no x-intercept.

- **y-intercept**: Set \( x = 0 \) and solve for \( f(0) \):

  \[
  f(0) = \frac{-3(0) + 15}{5(0) - 25} = \frac{15}{-25} = -0.6
  \]

  The y-intercept is \( (0, -0.6) \).

### Summary of Features

- **Domain**: \( (-\infty, 5) \cup (5, \infty) \)
- **Hole**: At \( (5, -0.6) \)
- **Vertical Asymptote**: None
- **Horizontal Asymptote**: \( y = -0.6 \)
- **x-intercept**: None (due to the hole at \( x = 5 \))
- **y-intercept**: \( (0, -0.6) \)

Would you like to explore any of these features further?

Here are some related questions:
1. How do you find oblique asymptotes for rational functions?
2. What happens to the graph of a function near a hole?
3. How can you determine if a function has a slant asymptote?
4. How does a hole differ from an x-intercept?
5. What are the implications of the horizontal asymptote on the behavior of the function as \( x \) approaches infinity?

**Tip:** Remember that holes in a function occur where a factor cancels out between the numerator and denominator, while vertical asymptotes occur where the denominator is zero but no cancellation occurs.

Explore the features of the function f(x) = (-3x + 15) / (5x - 25), including domain, holes, asymptotes, and intercepts. Understand how to determine where the function is defined, the presence of holes, and the behavior near asymptotes. Suitable for advanced high school students and college-level learners.

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