Final answer:
The domain of the function F(x) excludes any x-values that make the denominator zero. After factoring the denominator and solving for x, it is determined that the excluded value is x = 3. Thus, the domain of the function is all real numbers except for x = 3.
Step-by-step explanation:
The domain of a function includes all the possible values of x for which the function is defined. For the function F(x) = x + 6/(x^3 - x^2 - 9x + 9), the domain would exclude any x-values that cause the denominator to be zero, since division by zero is undefined. We need to factor the denominator to find these values.
The denominator can be factored as x^3 - x^2 - 9x + 9 = (x - 3)(x^2 + x - 3). Setting each factor to zero gives us the x-values that would result in division by zero:
x - 3 = 0 → x = 3
x^2 + x - 3 = 0 → This quadratic does not factor nicely, and we would need to use the quadratic formula to solve for x, but it's clear that x = 3 is already one of the roots. No other integer solutions exist from the quadratic.
Since the quadratic factor does not produce additional integer roots, the excluded value from the domain is x = 3. Therefore, the correct answer is option (b) x such that x ≠ 3.