The task is to determine the domain $D(f)$ and range $E(f)$ for each of the four graphs provided in the image. Let's analyze each graph:

1. Graph 1

• Domain $D(f)$: The graph exists for $x$ values from $-2$ to $4$. Hence, the domain is $D(f) = [-2, 4]$.
• Range $E(f)$: The graph covers $y$ values from $-3$ to $3$. Hence, the range is $E(f) = [-3, 3]$.

2. Graph 2

• Domain $D(f)$: The graph is defined for all $x$ values except at $x = 2$, where there is a vertical asymptote. Thus, the domain is $D(f) = (-\infty, 2) \cup (2, \infty)$.
• Range $E(f)$: The graph approaches all $y$ values except $y = 3$. Hence, the range is $E(f) = (-\infty, 3) \cup (3, \infty)$.

3. Graph 3

• Domain $D(f)$: The graph is defined for $x$ values from $-4$ to $2$. Hence, the domain is $D(f) = [-4, 2]$.
• Range $E(f)$: The graph covers $y$ values from $-2$ to $3$. Hence, the range is $E(f) = [-2, 3]$.

4. Graph 4

• Domain $D(f)$: The graph is defined for all $x$ values except at $x = 0$, where there is a vertical asymptote. Thus, the domain is $D(f) = (-\infty, 0) \cup (0, \infty)$.
• Range $E(f)$: The graph approaches all $y$ values except $y = -3$. Hence, the range is $E(f) = (-\infty, -3) \cup (-3, \infty)$.

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

Related Questions:

1. What is the significance of vertical asymptotes in determining the domain of a function?
2. How does the range of a function change when there is a horizontal asymptote?
3. Can a function have an unbounded domain but a bounded range? Provide an example.
4. How can piecewise functions affect the determination of the domain and range?
5. What are the key characteristics to look for when sketching a graph to determine its domain and range?

Tip:

When determining the domain and range from a graph, always pay attention to asymptotes and endpoints, as these indicate where the function does not exist or where it has restrictions.