https://file.aiquickdraw.com/mathbot/file/17245228748186qkg1xra.jpg

The expression given is:

\[
\frac{10x}{9x + 3}
\]

### Finding the Domain

The domain of a rational function is the set of all real numbers except where the denominator is equal to zero (since division by zero is undefined).

Set the denominator equal to zero and solve for \( x \):

\[
9x + 3 = 0
\]

Solving for \( x \):

\[
9x = -3
\]

\[
x = -\frac{1}{3}
\]

Thus, the domain is all real numbers except \( x = -\frac{1}{3} \). In interval notation, the domain is:

\[
(-\infty, -\frac{1}{3}) \cup (-\frac{1}{3}, \infty)
\]

### Finding the Range

To find the range, we analyze the behavior of the function as \( x \) approaches large positive and negative values and consider any restrictions that occur based on the form of the function.

1. **Horizontal Asymptote**: The degree of the polynomial in the numerator and denominator is the same (both are linear terms). Therefore, the horizontal asymptote is the ratio of the leading coefficients:
   \[
   \text{Horizontal Asymptote} = \frac{10}{9}
   \]
   This tells us that the function approaches \( y = \frac{10}{9} \) as \( x \to \pm \infty \).

2. **Critical Values**: Since the function is a rational function, it can take any value except at the vertical asymptote, which corresponds to the undefined point at \( x = -\frac{1}{3} \).

Thus, the range of the function is all real numbers except \( y = \frac{10}{9} \), which the function approaches but never actually reaches. In interval notation, the range is:

\[
(-\infty, \frac{10}{9}) \cup (\frac{10}{9}, \infty)
\]

### Final Answer

- **Domain**: \( (-\infty, -\frac{1}{3}) \cup (-\frac{1}{3}, \infty) \)
- **Range**: \( (-\infty, \frac{10}{9}) \cup (\frac{10}{9}, \infty) \)

Would you like more details on this, or do you have any questions?

---

Here are 5 related questions to explore:
1. How do vertical and horizontal asymptotes influence the shape of the graph of a rational function?
2. Can a rational function have more than one vertical or horizontal asymptote?
3. How do you determine the slant asymptote of a rational function when the degrees of the numerator and denominator differ?
4. What happens to the range of a rational function when the numerator has a higher degree than the denominator?
5. How do holes (removable discontinuities) affect the domain and range of rational functions?

**Tip:** When finding the domain of a rational function, always first check for any values that make the denominator zero, as these will exclude values from the domain.

Explore how to find the domain and range of the rational function \( \frac{10x}{9x + 3} \). Learn about horizontal asymptotes, domain restrictions, and the range of the function. Suitable for high school students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation