The function $Y = \frac{4x^2 + 3}{5x^2 + 4}$ is a rational function. Let's analyze it further:

Key Features and Analysis

1. Domain:

   • The domain of a rational function excludes any values of $x$ that make the denominator zero.
   • Solve $5x^2 + 4 = 0$: $x^2 = -\frac{4}{5}$ Since there are no real solutions (negative inside the square root), the denominator never becomes zero. Therefore, the domain is all real numbers $\mathbb{R}$.

2. Intercepts:

   • Y-intercept: Substitute $x = 0$: $Y = \frac{4(0)^2 + 3}{5(0)^2 + 4} = \frac{3}{4}$ The y-intercept is $\frac{3}{4}$.
   • X-intercept: Solve $4x^2 + 3 = 0$: $x^2 = -\frac{3}{4}$ No real solutions exist, so there are no x-intercepts.

3. Asymptotes:

   • Horizontal Asymptote: Compare the degrees of the numerator and denominator: The degrees of both numerator and denominator are equal (both are quadratic). Therefore, the horizontal asymptote is determined by the ratio of the leading coefficients: $y = \frac{4}{5}$
   • Vertical Asymptote: Since the denominator has no real roots, there are no vertical asymptotes.

4. Behavior:

   • As $x \to \infty$ or $x \to -\infty$, the function approaches the horizontal asymptote $y = \frac{4}{5}$.

Summary

• Domain: $(-\infty, \infty)$
• Y-intercept: $\frac{3}{4}$
• X-intercept: None
• Horizontal Asymptote: $y = \frac{4}{5}$
• Vertical Asymptote: None

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5 Related Questions

1. How do you find the intercepts of more complex rational functions?
2. What happens if the denominator has real roots? How does the domain change?
3. Can a rational function have both horizontal and oblique asymptotes?
4. How do you calculate slant asymptotes for rational functions?
5. What is the significance of degrees in the numerator and denominator of a rational function?

Tip: Always check for undefined values in the denominator when