The given function is $f(x) = -(x + 3)(x - 1)$, and its graph is a downward-facing parabola. Let’s analyze the domain and range based on the provided graph.

Domain:

• The domain refers to all possible $x$-values for which the function is defined. From the graph, the parabola extends infinitely to the left and right, meaning the function is defined for all real numbers.
  • Hence, the domain is $(-\infty, \infty)$, or simply all real numbers.

Range:

• The range refers to all possible $y$-values the function can take. The vertex of the parabola is at its maximum point, which is around $y = 4$. The parabola opens downward, so the function’s values are less than or equal to this maximum value.
  • Hence, the range is $(-\infty, 4]$.

Correct Answer:

The correct choice is the one that reflects that the domain is all real numbers and the range is all real numbers less than or equal to 4, which corresponds to:

• Option C: The domain is all real numbers, and the range is all real numbers less than or equal to 4.

Would you like further clarification or additional details on how the domain and range are determined?

Here are 5 related questions to explore further:

1. What are some common features of parabolas and how do they affect domain and range?
2. How does the sign of the leading coefficient in a quadratic function affect the graph's direction?
3. What is the significance of the vertex in determining the range of a quadratic function?
4. Can a quadratic function ever have a domain that is not all real numbers?
5. How would the domain and range change if the function had been shifted vertically or horizontally?

Tip: The domain of a quadratic function is almost always all real numbers, but the range depends on the direction of the parabola and its vertex.