Final answer:
Discontinuities occur when the denominator of a rational function is zero. For the function f(x) = (x^2 + 8x + 4) / (x^2 - x - 6), the discontinuities occur at x = -2 and x = 3, as these values would make the denominator, and thus the function itself, undefined.
Step-by-step explanation:
The question asks for the value at which the function f(x) = \frac{x^2 + 8x + 4}{x^2 - x - 6} is discontinuous. To find the discontinuities of a rational function, we need to determine where the denominator equals zero, because division by zero is undefined. The denominator is x^2 - x - 6, which is a quadratic equation. To find its roots, we can factor it or use the quadratic formula. Factoring x^2 - x - 6, we find that (x - 3)(x + 2) = 0. Therefore, the roots of the equation are x = 3 and x = -2.
To determine which value is a discontinuity, we simply check which of the roots causes the denominator to be zero, thereby making the function undefined. As we have already found the roots, we know that both of these values will render the function undefined.
Thus, the possible values for discontinuity in the given function are both B) -2 and C) 3. The question appears to ask for a single value, but based on the given function, there are indeed two values where the function is discontinuous.