https://file.aiquickdraw.com/mathbot/file/1725785220627sli1xrdn.jpg

The function given in the image is:

\[
R(x) = \frac{x - 1}{3x^2 + 3x}
\]

We are asked to find the following:
a) Domain  
b) x-intercepts  
c) y-intercepts  
d) Zeros  
e) Vertical asymptote  
f) Horizontal asymptote  
g) Hole

Let’s go step by step:

### a) Domain:
The domain of a rational function is all real numbers except where the denominator is equal to zero.

The denominator is \( 3x^2 + 3x = 3x(x + 1) \).

Setting the denominator equal to zero and solving for \( x \):
\[
3x(x + 1) = 0
\]
\[
x = 0 \quad \text{or} \quad x = -1
\]

Thus, the domain is all real numbers except \( x = 0 \) and \( x = -1 \).

**Domain**: \( (-\infty, -1) \cup (-1, 0) \cup (0, \infty) \)

### b) x-intercepts:
The x-intercepts occur when \( R(x) = 0 \), which means the numerator is equal to zero.

The numerator is \( x - 1 \). Set it equal to zero:
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

So, the x-intercept is at \( x = 1 \).

### c) y-intercepts:
The y-intercept occurs when \( x = 0 \).

However, in this case, \( x = 0 \) is not in the domain of the function, so there is no y-intercept.

**y-intercept**: None

### d) Zeros:
The zeros of the function are the same as the x-intercepts. Therefore, the zero is at \( x = 1 \).

**Zero**: \( x = 1 \)

### e) Vertical Asymptote:
Vertical asymptotes occur where the denominator equals zero, provided the numerator is non-zero.

From the domain calculation, the denominator is zero at \( x = 0 \) and \( x = -1 \). Let's check if the numerator is non-zero at these points:

- At \( x = 0 \), the numerator \( x - 1 = -1 \), so there is a vertical asymptote at \( x = 0 \).
- At \( x = -1 \), the numerator \( x - 1 = -2 \), so there is a vertical asymptote at \( x = -1 \).

**Vertical asymptotes**: \( x = 0 \) and \( x = -1 \)

### f) Horizontal Asymptote:
For horizontal asymptotes, we compare the degrees of the numerator and the denominator.

- The degree of the numerator is 1.
- The degree of the denominator is 2.

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \( y = 0 \).

**Horizontal asymptote**: \( y = 0 \)

### g) Hole:
A hole occurs where a factor cancels from both the numerator and the denominator.

In this case, there are no common factors between the numerator and denominator, so there is no hole.

**Hole**: None

---

Would you like further details or clarifications?  
Here are five related questions you can explore:

1. How do you determine the slant asymptote for a rational function?
2. What happens to the graph of a function at a vertical asymptote?
3. How can you determine end behavior for different rational functions?
4. What is the relationship between a function's degree and its asymptotes?
5. How do you calculate the limits approaching vertical asymptotes?

**Tip:** Always check if common factors can cancel when finding vertical asymptotes and holes in rational functions.

Find the domain, x-intercepts, y-intercepts, zeros, vertical asymptote, horizontal asymptote, and hole of the function R(x) = (x - 1) / (3x^2 + 3x).

Learn how to analyze the rational function R(x) = (x - 1) / (3x^2 + 3x) by finding its domain, intercepts, zeros, vertical and horizontal asymptotes, and holes. This guide covers key concepts and formulas essential for understanding rational functions, suitable for students in Grades 9-12.

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