Let's analyze the graph provided and address each of the questions:

(a) Domain and Range

• Domain: The function has a vertical asymptote at $x = 0$, indicating that it is undefined at $x = 0$. Therefore, the domain is $x \in (-\infty, 0) \cup (0, \infty)$.
• Range: The function values extend from $y = -\infty$ up to $y = \infty$ as shown by the graph. There are no restrictions on $y$, so the range is $y \in (-\infty, \infty)$.

(b) Intercepts

• x-intercept: From the graph, the function crosses the x-axis at $x = 4$. Thus, the x-intercept is $(4, 0)$.
• y-intercept: The function does not cross the y-axis due to the vertical asymptote at $x = 0$. Therefore, there is no y-intercept.

(c) Horizontal Asymptotes

• There are no horizontal asymptotes in this graph. As $x \to \infty$, the function goes to $y = \infty$, and as $x \to -\infty$, the function goes to $y = -\infty$.

(d) Vertical Asymptotes

• The graph shows a vertical asymptote at $x = 0$. This is evident from the behavior of the function, as it tends towards $\infty$ as $x$ approaches 0 from the right and towards $-\infty$ as $x$ approaches 0 from the left.

(e) Oblique Asymptotes

• The graph has an oblique (or slant) asymptote. Observing the graph's behavior as $x \to \infty$, it appears to approach a line with a positive slope, indicating an oblique asymptote.

Summary:

1. Domain: $x \in (-\infty, 0) \cup (0, \infty)$
2. Range: $y \in (-\infty, \infty)$
3. x-intercept: $(4, 0)$; no y-intercept.
4. Vertical Asymptote: $x = 0$
5. No Horizontal Asymptote
6. Oblique Asymptote: Exists as $x \to \infty$.

Would you like further details on any part?

Related Questions:

1. How do you determine asymptotes from a function’s equation?
2. What other types of functions might exhibit similar asymptotic behavior?
3. How can we algebraically confirm the oblique asymptote if given an equation?
4. Why does the function have no horizontal asymptote?
5. How does the presence of a vertical asymptote affect the domain of a function?

Tip:

When analyzing a function graph, always check for symmetry, as it can provide insight into intercepts and asymptotes.