The function you've provided is:

\[
f(x) = \frac{4x^2}{9x^2 - 25}
\]

Let's analyze this function by looking at several key features:

### 1. **Domain:**
The denominator \( 9x^2 - 25 \) cannot be zero, as division by zero is undefined. To find the points where the denominator is zero:

\[
9x^2 - 25 = 0
\]
\[
9x^2 = 25
\]
\[
x^2 = \frac{25}{9}
\]
\[
x = \pm \frac{5}{3}
\]

Thus, the domain excludes \( x = \pm \frac{5}{3} \).

So, the domain is:
\[
x \in (-\infty, -\frac{5}{3}) \cup (-\frac{5}{3}, \frac{5}{3}) \cup (\frac{5}{3}, \infty)
\]

### 2. **Vertical Asymptotes:**
The function has vertical asymptotes at the points where the denominator equals zero. Therefore, vertical asymptotes occur at:
\[
x = \pm \frac{5}{3}
\]

### 3. **Horizontal Asymptote:**
To find the horizontal asymptote, consider the degree of the numerator and the denominator. Since both the numerator and the denominator are quadratic functions (degree 2), we can compare the leading coefficients:

\[
\lim_{x \to \infty} f(x) = \frac{4x^2}{9x^2} = \frac{4}{9}
\]

Thus, the horizontal asymptote is:
\[
y = \frac{4}{9}
\]

### 4. **Intercepts:**
- **x-intercept**: Set \( f(x) = 0 \), which implies \( 4x^2 = 0 \). So, \( x = 0 \).
- **y-intercept**: The y-intercept occurs when \( x = 0 \), which is also \( f(0) = 0 \).

Would you like more details or have any questions? Here are some related questions:

1. How do you graph rational functions like this one?
2. What happens to \( f(x) \) near the vertical asymptotes?
3. What role does the leading coefficient play in determining asymptotes?
4. How do we find the behavior of a function as \( x \to 0 \)?
5. How do we simplify functions before analyzing them?

**Tip:** Always check for factorable forms in the denominator or numerator to simplify rational functions!

Explore the detailed analysis of the rational function f(x) = 4x^2 / (9x^2 - 25), covering domain restrictions, vertical and horizontal asymptotes, and intercepts. Learn key algebraic techniques for analyzing rational functions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation