Let's examine the function provided in the image and identify the key features requested:

Given: $f(x) = \frac{x + 2}{x^2 - 3x - 10}$

Step-by-Step Analysis

1. Domain: The domain excludes values that make the denominator zero. Set the denominator equal to zero and solve for $x$: $x^2 - 3x - 10 = 0$ Factoring gives: $(x - 5)(x + 2) = 0$ So, $x = 5$ and $x = -2$ are values that make the denominator zero. Thus, the domain is all real numbers except $x = 5$ and $x = -2$.

2. Range: The range is a bit more complex to determine without graphing. For rational functions, you can generally find the range by considering the behavior as $x$ approaches the vertical asymptotes, horizontal asymptotes, or any holes. I’ll note that the exact range requires analysis or graphing, often using limits.

3. x-intercept(s): Set the numerator equal to zero to find the x-intercepts: $x + 2 = 0 \Rightarrow x = -2$ However, $x = -2$ is also a zero of the denominator, meaning there's a hole at $x = -2$, not an x-intercept.

4. y-intercept: Substitute $x = 0$ into $f(x)$: $f(0) = \frac{0 + 2}{0^2 - 3(0) - 10} = \frac{2}{-10} = -\frac{1}{5}$ So, the y-intercept is $\left(0, -\frac{1}{5}\right)$.

5. Vertical Asymptote: Vertical asymptotes occur at values of $x$ that make the denominator zero but not the numerator. Since $x = 5$ is not a zero of the numerator, there is a vertical asymptote at $x = 5$.

6. Horizontal Asymptote: Since the degree of the denominator (2) is higher than the degree of the numerator (1), the horizontal asymptote is $y = 0$.

7. Hole: There is a hole at $x = -2$ because both the numerator and denominator are zero at this point. To find the y-coordinate of the hole, simplify $f(x)$ near $x = -2$: $f(x) = \frac{x + 2}{(x - 5)(x + 2)} = \frac{1}{x - 5} \text{ (for } x \neq -2\text{)}$ Plugging in $x = -2$ into the simplified form gives: $y = \frac{1}{-2 - 5} = -\frac{1}{7}$ So, there is a hole at $\left(-2, -\frac{1}{7}\right)$.

8. End Behavior: As $x \to \pm \infty$, the function approaches the horizontal asymptote $y = 0$.

Let me know if you’d like further details or explanations on any specific part of this analysis.

Additional Questions to Explore:

1. How do we determine the exact range of this function?
2. Why does the degree of the numerator and denominator affect the horizontal asymptote?
3. Can there be more than one vertical asymptote in rational functions? Why or why not?
4. How would you graph a hole in a function like this?
5. What does the end behavior tell us about the function as $x \to \infty$ or $x \to -\infty$?

Tip:

When analyzing rational functions, always start by factoring both the numerator and the denominator fully to simplify finding intercepts, asymptotes, and holes.