Final answer:
The value of a that ensures the function f(x) is continuous for all real numbers x is 4. This is because it cancels the potential discontinuity at x = 0, making f(x) continuous over the entire real number line. Hence, the correct answer is D. (4) only.
Step-by-step explanation:
The continuity of the function f(x) = frac{(x-1)(x^2-4)}{x^2-a} depends on the values that do not cause a division by zero in the denominator. The function is not defined at x = 0 because that would make the denominator zero. However, the factors (x-1)(x+2)(x-2) in the numerator indicate potential zeros that, if equal to a, would cancel out the discontinuity at x = 0.
To ensure continuity for all x, we need to have a equal to the product of (potentially zero) terms of the numerator when x = 0. However, at x = 0, (x-1)(x+2)(x-2) equals (0-1)(0+2)(0-2), which simplifies to -4. Hene, for a = 4, the term (x^2-4) in the numerator cancels out with the x^2-4 in the denominator when x = 0, making f(x) continuous for all real numbers.
Therefore, the correct answer for the value of a that makes f(x) continuous for all real numbers (x) is D. (4) only.