Final answer:
The equations C, D, E, and F could define the function f, as they are variations of quadratic functions in either standard or factored form, representing possible functions for f.
Step-by-step explanation:
We are tasked with selecting equations that could define the function f. Notice that original functions and those that result from replacing x with -x are different forms of the same underlying function when they have the same structure and different signs on the terms that involve x.
- Equation A: f(-x) = -x² + 8x - 12 represents a function where every x is replaced with -x and the terms have different signs compared to the standard form of quadratic equations, which means it is a reflection over the y-axis of an unlisted equation in its standard form.
- Equation B: f(-x) = x² - 8x + 12 also is a function where x has been replaced with -x but retains the same structure as a standard quadratic, indicating it is not the same as f(x).
- Equation C: f(x) = (x + 2)(x + 6) is in factored form indicating that it is a quadratic function, which could define f.
- Equation D: f(x) = (x - 4)² + 4 is another form of a quadratic function, representing a parabola with vertex at (4, 4) which could represent f.
- Equation E: f(x) = (x - 4)² - 4 is similar to Equation D but with the vertex at (4, -4), so it could also represent f.
- Equation F: f(x) = (x - 2)(x - 6) is a factored quadratic function, much like Equation C, and could also represent f.
Therefore, the equations that could define the function f are C, D, E, and F.