The problem asks to find the range of the function $f(x) = x^2 - 2x - 7$ given the domain $\{-5, -1, 4\}$.

To find the range, we'll evaluate the function for each value in the domain.

1. For $x = -5$: $f(-5) = (-5)^2 - 2(-5) - 7 = 25 + 10 - 7 = 28$

2. For $x = -1$: $f(-1) = (-1)^2 - 2(-1) - 7 = 1 + 2 - 7 = -4$

3. For $x = 4$: $f(4) = (4)^2 - 2(4) - 7 = 16 - 8 - 7 = 1$

Thus, the range of the function for the domain $\{-5, -1, 4\}$ is $\{28, -4, 1\}$.

The correct answer is A: {-4, 1, 28}.

Would you like further details on any of the calculations?

Here are five related questions to deepen your understanding:

1. How would the range change if the domain were extended?
2. What is the vertex of the parabola $f(x) = x^2 - 2x - 7$?
3. How would you determine the range for an arbitrary quadratic function?
4. What role does the axis of symmetry play in the range of a quadratic function?
5. How can you find the minimum or maximum value of a quadratic function using derivatives?

Tip: When working with quadratic functions, identifying the vertex can help you quickly determine the range when the domain includes all real numbers.