Final answer:
The domain of the function F(x) is all real numbers except for x = 0, x = 1, and x = -1, which are excluded because they would cause division by zero within the function.
Step-by-step explanation:
The student has asked to find the domain of the function F(x) = \frac{(x+1)(x^3-x)}{(x^2+2)}.
To find the domain of this function, we need to identify the values of x for which the function is defined. The domain of a function is the set of all possible inputs (in this case, values of x) for which the function produces a real number as an output.
In the given function, we have a denominator of x^2+2, and because we cannot divide by zero, we need to ensure that the denominator never equals zero. Since x^2 is always positive for all real x, and 2 is a positive number, x^2+2 will never be zero. Therefore, there are no restrictions on x based on the denominator.
However, we also have a fractional exponent (x^3 - x)^{-1} which implies division by this term. The base of this exponent (x^3 - x) cannot be zero, as that would result in division by zero. If we set x^3 - x equal to zero, we get x(x^2 - 1) = 0, which gives us x = 0, x = 1, and x = -1 as the roots. These are the values of x that cause this term to be zero, so they must be excluded from the domain.
Therefore, the domain of the function F(x) is all real numbers except x = 0, x = 1, and x = -1.