Let's analyze each graph to determine the domain, range, and whether it is the graph of a function by applying the vertical line test.

(a) Graph

• Domain: All real numbers $(-\infty, \infty)$
• Range: All real numbers $(-\infty, \infty)$
• Function: Yes, this graph is a function because it passes the vertical line test.

(b) Graph

• Domain: $x = 2$ (the graph only exists at this x-value)
• Range: All real numbers $(-\infty, \infty)$
• Function: No, this is not a function. A vertical line at $x = 2$ fails the vertical line test.

(c) Graph

• Domain: $x \in [-3, 3]$ (the circle spans from $x = -3$ to $x = 3$)
• Range: $y \in [-3, 3]$ (the circle spans from $y = -3$ to $y = 3$)
• Function: No, this is not a function. A vertical line crosses the circle at two points.

(d) Graph

• Domain: All real numbers $(-\infty, \infty)$
• Range: $y \in (-\infty, 0]$ (the graph does not go above the x-axis)
• Function: Yes, it passes the vertical line test.

(e) Graph

• Domain: $x = -5$ (the graph only exists at this x-value)
• Range: All real numbers $(-\infty, \infty)$
• Function: No, this is not a function due to a vertical line at $x = -5$.

(f) Graph

• Domain: All real numbers $(-\infty, \infty)$
• Range: $y \in [1, \infty)$
• Function: Yes, it passes the vertical line test.

(g) Graph

• Domain: All real numbers $(-\infty, \infty)$
• Range: $y \in (-\infty, 4]$
• Function: Yes, it passes the vertical line test.

(h) Graph

• Domain: $x \geq -5$ (the graph only exists for values greater than or equal to $x = -5$)
• Range: All real numbers $(-\infty, \infty)$
• Function: No, it fails the vertical line test.

(i) Graph

• Domain: $x \in (-\infty, 1) \cup (1, \infty)$ (there is an asymptote at $x = 1$)
• Range: All real numbers $(-\infty, \infty)$
• Function: Yes, it passes the vertical line test.

Would you like further details on any specific graph or additional explanations?

Related Questions:

1. What is the vertical line test, and how does it help identify functions?
2. How do we find the domain and range of a graph?
3. Can a graph with a vertical line be a function in any case?
4. How do asymptotes affect the domain of a function?
5. What are the characteristics of polynomial functions based on their graphs?

Tip: The domain of a function is the set of all possible x-values (inputs), while the range is the set of all possible y-values (outputs).