https://file.aiquickdraw.com/mathbot/file/1726063162389dw1x9ay8.jpg

The image contains two rational functions for which we need to determine the domain, range, and asymptotes.

### Function 1: \( y = \frac{2x - 5}{2x + 5} \)
1. **Domain**: The domain of a rational function is all real numbers except where the denominator is zero.
   \[
   2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2}
   \]
   So, the domain is \( x \in \mathbb{R} \setminus \left\{ -\frac{5}{2} \right\} \).

2. **Range**: To find the range, we can use the fact that horizontal asymptotes can give insight into the behavior of the function as \( x \to \pm \infty \). Since the degrees of the numerator and denominator are the same, the horizontal asymptote is determined by the ratio of the leading coefficients:
   \[
   \text{Horizontal Asymptote: } y = \frac{2}{2} = 1
   \]
   However, since the function never actually reaches the horizontal asymptote but can approach any value above or below 1, the range is \( y \in \mathbb{R} \setminus \{ 1 \} \).

3. **Asymptotes**:
   - **Vertical Asymptote**: Occurs where the denominator is zero, which is \( x = -\frac{5}{2} \).
   - **Horizontal Asymptote**: \( y = 1 \).

---

### Function 2: \( y = \frac{x^2 - 1}{2x^2 - 3x - 2} \)
1. **Domain**: We need to find where the denominator equals zero to exclude these values from the domain.
   \[
   2x^2 - 3x - 2 = 0
   \]
   Solving this quadratic equation using the quadratic formula:
   \[
   x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-2)}}{2(2)} = \frac{3 \pm \sqrt{9 + 16}}{4} = \frac{3 \pm \sqrt{25}}{4}
   \]
   \[
   x = \frac{3 \pm 5}{4} \quad \Rightarrow \quad x = 2 \quad \text{or} \quad x = -\frac{1}{2}
   \]
   So, the domain is \( x \in \mathbb{R} \setminus \{ 2, -\frac{1}{2} \} \).

2. **Range**: Finding the range analytically for more complex rational functions is challenging, but we can use a numerical or graphical approach. Generally, it involves investigating the behavior of the function as \( x \to \pm \infty \).

3. **Asymptotes**:
   - **Vertical Asymptotes**: These occur where the denominator is zero, which gives vertical asymptotes at \( x = 2 \) and \( x = -\frac{1}{2} \).
   - **Horizontal Asymptote**: Since the degrees of the numerator and denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients:
     \[
     y = \frac{1}{2}
     \]

---

### Summary:
- **For \( y = \frac{2x - 5}{2x + 5} \)**:
  - Domain: \( x \in \mathbb{R} \setminus \left\{ -\frac{5}{2} \right\} \)
  - Range: \( y \in \mathbb{R} \setminus \{ 1 \} \)
  - Vertical Asymptote: \( x = -\frac{5}{2} \)
  - Horizontal Asymptote: \( y = 1 \)

- **For \( y = \frac{x^2 - 1}{2x^2 - 3x - 2} \)**:
  - Domain: \( x \in \mathbb{R} \setminus \{ 2, -\frac{1}{2} \} \)
  - Range: Needs further analysis or graphical investigation.
  - Vertical Asymptotes: \( x = 2 \) and \( x = -\frac{1}{2} \)
  - Horizontal Asymptote: \( y = \frac{1}{2} \)

Let me know if you'd like further details or a graphical analysis!

Here are five related questions to expand your understanding:
1. How do we determine slant (oblique) asymptotes for rational functions?
2. What methods can be used to find the range of a rational function besides graphical approaches?
3. How do horizontal asymptotes affect the long-term behavior of a rational function?
4. What are removable discontinuities in rational functions, and how can they be identified?
5. How does the degree of the numerator and denominator influence the behavior of a rational function?

**Tip**: Always check for common factors in the numerator and denominator before determining vertical asymptotes. Sometimes a factor can cancel, resulting in a removable discontinuity instead of an asymptote.

Determine the domain, range, and asymptotes for the functions: y = (2x - 5) / (2x + 5) and y = (x^2 - 1) / (2x^2 - 3x - 2).

Learn how to determine the domain, range, and asymptotes of the rational functions y = (2x - 5) / (2x + 5) and y = (x^2 - 1) / (2x^2 - 3x - 2). This task covers key algebraic concepts such as rational functions, domain restrictions, and asymptote behavior. Suitable for advanced high school students in Grades 11-12.

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