The image contains two rational functions for which you need to determine the domain, range, and asymptotes. Here are the functions:

1. $y = \frac{2x - 5}{2x + 5}$
2. $y = \frac{x^2 - 1}{2x^2 - 3x - 2}$

Let's break them down one by one.

1. $y = \frac{2x - 5}{2x + 5}$

• Domain: The domain is all real numbers except where the denominator is zero. Solve $2x + 5 = 0$: $2x + 5 = 0 \implies x = -\frac{5}{2}$ So, the domain is $x \in \mathbb{R}$, $x \neq -\frac{5}{2}$.

• Vertical Asymptote: A vertical asymptote occurs where the denominator is zero. In this case, there is a vertical asymptote at $x = -\frac{5}{2}$.

• Horizontal Asymptote: Compare the degrees of the numerator and the denominator. Since both are linear (degree 1), the horizontal asymptote is determined by the ratio of the leading coefficients: $y = \frac{2}{2} = 1$ Thus, the horizontal asymptote is $y = 1$.

• Range: The range of the function excludes the horizontal asymptote $y = 1$, so the range is $y \in \mathbb{R}, y \neq 1$.

2. $y = \frac{x^2 - 1}{2x^2 - 3x - 2}$

• Domain: The domain excludes values where the denominator is zero. Factor the denominator: $2x^2 - 3x - 2 = (2x + 1)(x - 2)$ So, the domain is all real numbers except $x = -\frac{1}{2}$ and $x = 2$.

• Vertical Asymptotes: Vertical asymptotes occur where the denominator is zero. From the factored form, we get vertical asymptotes at: $x = -\frac{1}{2}, \quad x = 2$

• Horizontal Asymptote: Since the degrees of the numerator and denominator are the same (degree 2), the horizontal asymptote is determined by the ratio of the leading coefficients: $y = \frac{1}{2}$ So, the horizontal asymptote is $y = \frac{1}{2}$.

• Range: The range excludes the horizontal asymptote $y = \frac{1}{2}$, so the range is $y \in \mathbb{R}, y \neq \frac{1}{2}$.

Would you like more details or explanations on any of these points?

Related Questions:

1. How do you determine slant (oblique) asymptotes for rational functions?
2. What is the general method to find the domain of any rational function?
3. Can a rational function have both a horizontal and slant asymptote?
4. What are the conditions for a rational function to have no horizontal asymptote?
5. How do you determine the behavior of a function near its vertical asymptote?

Tip: Always check if a rational function can be simplified before analyzing its asymptotes and other characteristics.