The function f(x) = x^2 - 4 has which domain and range?

• A. Domain: all real numbers; Range: y ≥ -4.
• B. Domain: x ≥ 0; Range: y ≥ 0.
• C. Domain: all real numbers; Range: y ≤ -4.
• D. Domain: all real numbers; Range: y ≤ 4.

Answer: (A)

A quadratic like f(x) = x^2 - 4 has domain all real numbers because you can plug any real x into x^2 and still subtract 4 without any restrictions. The graph is a parabola opening upward and shifted down by 4, so its lowest point is at the vertex (0, -4). Since x^2 is always nonnegative, f(x) = x^2 - 4 is always at least -4, with values increasing without bound as |x| grows. So the range is y ≥ -4.

That’s why the description with domain all real numbers and range y ≥ -4 fits. The other options would require either restricting x to nonnegative values, which would only show part of the parabola, or implying a maximum value like ≤ -4 or ≤ 4, which doesn’t match an upward-opening parabola that has no upper bound.