To determine the domain and range of the function represented by the graph, let's break it down:

Given:

• The graph represents an upward-opening parabola with a vertex at $(1, -16)$.
• The parabola intersects the X-axis at points $(-3, 0)$ and $(5, 0)$.

Domain:

• The domain of a function is the set of all possible values of $x$ for which the function is defined.
• Since the parabola opens upward and extends infinitely to the left and right, the function is defined for all real numbers.
• Thus, the domain is $\textbf{(-∞, ∞)}$.

Range:

• The range of a function is the set of all possible values of $y$ that the function can take.
• The vertex of the parabola is at $(1, -16)$, and since the parabola opens upward, the smallest value of $y$ is $-16$.
• The function takes all values of $y$ greater than or equal to $-16$.
• Thus, the range is $\textbf{[-16, ∞)}$.

Correct Responses:

1. Domain: $\textbf{(-∞, ∞)}$
2. Range: $\textbf{[-16, ∞)}$

These are the correct choices for the domain and range.

Do you have any questions or need further details?

Related Questions:

1. How do you determine the domain of more complex functions?
2. What does it mean for a parabola to have a domain of $(-∞, ∞)$?
3. How can you find the range of a function without graphing it?
4. What are the implications of a parabola having a minimum or maximum value?
5. How do vertical shifts affect the range of a parabola?

Tip: The domain of a parabola that opens up or down is always $(-∞, ∞)$, but its range depends on the vertex.