The function given in the image is $f(x) = \frac{2}{x} - 5$. Let's analyze the function to answer the following:

Step-by-step analysis:

1. Domain: The domain is determined by the values of $x$ for which the function is defined. Since division by zero is undefined, the function will have a restriction where $x = 0$. Therefore, the domain is all real numbers except $x = 0$, which can be written as: $\text{Domain}: (-\infty, 0) \cup (0, \infty)$ This excludes 0 from the domain.

2. Vertical Asymptote (VA): Vertical asymptotes occur where the denominator of a rational function is 0. In this case, the denominator $x = 0$ leads to a vertical asymptote at $x = 0$.

3. Horizontal Asymptote (HA): Horizontal asymptotes are determined by analyzing the behavior of the function as $x \to \infty$ or $x \to -\infty$.

   • As $x \to \infty$, the $\frac{2}{x} \to 0$, so $f(x) \to -5$. Therefore, the horizontal asymptote is $y = -5$.

4. Range: The range is the set of values $f(x)$ can take. Since the horizontal asymptote is $y = -5$ and the function never actually equals $-5$, but can take values close to it or much greater/less than it, the range is: $\text{Range}: (-\infty, -5) \cup (-5, \infty)$

Filling in the blanks:

• Domain: The correct value is $(-\infty, 0) \cup (0, \infty)$. It seems the provided numbers are 18 and 24, but these do not align with the domain.
• Vertical Asymptote (VA): The correct vertical asymptote is $x = 0$. There is a value of 16, but this doesn't seem appropriate for a vertical asymptote.
• Horizontal Asymptote (HA): The correct horizontal asymptote is $y = -5$. The value of 4 is present, but it should be $y = -5$.
• Range: The range should be $(-\infty, -5) \cup (-5, \infty)$, not the provided values of 24 and 18.

Would you like more details on any of these concepts? Here are five related questions to expand on this:

1. How do you determine vertical and horizontal asymptotes for other rational functions?
2. What happens to the function as $x \to 0^+$ and $x \to 0^-$?
3. Can a rational function have more than one horizontal asymptote?
4. How does the transformation $f(x) = \frac{a}{x} - b$ affect the graph of $f(x) = \frac{1}{x}$?
5. How do you find the x- and y-intercepts of rational functions like this one?

Tip: To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the denominator’s degree is higher, the horizontal asymptote is $y = 0$, but if they are the same degree, it's the ratio of the leading coefficients.