To determine the domain of the function based on the provided information about the parabola, we analyze the features given:
- The parabola has its vertex at (0, 4).
- It passes through the x-axis at (-2, 0) and (2, 0).
This indicates that the parabola opens downwards because the vertex is at the highest point (4), and it intersects the x-axis at (-2, 0) and (2, 0).
The x-values where the parabola intersects the x-axis, namely -2 and 2, suggest that the function has defined values for all x in the interval from -2 to 2 (both inclusive) since a polynomial function, like a quadratic, is defined for every real number.
Thus, the domain of the function is: \[ -2 \leq x \leq 2 \]
However, based on your options and the underlined inequalities implying inclusivity, the choice that represents this correctly would be limited to the choices provided:
- A. -2 < x < 2: Incorrect, as it does not include the endpoints.
- B. 0 < x < 2: Incorrect, as it does not include the negative side and the domain includes -2.
- C. x is all real numbers less than or equal to 4: Incorrect, as the domain is not restricted to y-values.
- D. x is all real numbers: Incorrect, as not all real numbers are part of the domain given the intercepts.
Since none of the options accurately capture the full range of the domain as derived from the graph description, we summarize that the correct answer should ideally convey the inclusion of the endpoints from -2 to 2. In scenarios like this, if you must choose, opt for the answer that comes closest, understanding the graph to fall within the defined range of the x-values.