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Let's solve the problem step by step for the rational function \( g(x) = \frac{3x}{2x + 1} \):

### 1. **Domain**:
The domain of a rational function is all real numbers except where the denominator is zero.

We set the denominator equal to zero to find the restriction:
\[
2x + 1 = 0
\]
Solving for \(x\):
\[
x = -\frac{1}{2}
\]

Thus, the domain is:
\[
x \in (-\infty, -\frac{1}{2}) \cup (-\frac{1}{2}, \infty)
\]

### 2. **x-intercept**:
The x-intercept occurs when \(g(x) = 0\). For a rational function, this happens when the numerator is zero (as long as the denominator is not zero).

Setting the numerator \(3x = 0\), we find:
\[
x = 0
\]

Thus, the x-intercept is at:
\[
(0, 0)
\]

### 3. **y-intercept**:
The y-intercept occurs when \(x = 0\). Substituting \(x = 0\) into the function \(g(x)\):

\[
g(0) = \frac{3(0)}{2(0) + 1} = 0
\]

Thus, the y-intercept is also at:
\[
(0, 0)
\]

### 4. **Vertical Asymptote**:
Vertical asymptotes occur where the denominator is zero, and the function is undefined.

From the earlier calculation, the denominator is zero when \(x = -\frac{1}{2}\).

Thus, the vertical asymptote is at:
\[
x = -\frac{1}{2}
\]

### 5. **Horizontal Asymptote**:
For rational functions, the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator.

- The degree of the numerator (3x) is 1.
- The degree of the denominator (2x + 1) is also 1.

When the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients. In this case, the leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2.

Thus, the horizontal asymptote is:
\[
y = \frac{3}{2}
\]

---

Would you like further details or have any questions on any of these points? Here are 5 related questions for practice:

1. How would the function change if the numerator was \(4x\) instead of \(3x\)?
2. What would be the effect on the horizontal asymptote if the degrees of the numerator and denominator were different?
3. Can a rational function have multiple vertical asymptotes?
4. How does the graph of the function behave near the vertical asymptote?
5. What happens to the graph as \(x\) approaches positive or negative infinity?

**Tip:** Always check both the numerator and denominator separately when solving for intercepts and asymptotes in rational functions.

Find the domain, x-intercept, y-intercept, vertical asymptote, and horizontal asymptote of the rational function g(x) = 3x / (2x + 1).

Find the domain, x-intercept, y-intercept, vertical asymptote, and horizontal asymptote of the rational function g(x) = 3x / (2x + 1). This problem involves key concepts of rational functions and asymptotic behavior, suitable for students in Grades 9-12.

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