Before solving, it is important to know if the x^3+3x is ENTIRELY in the denominator, or if the function F is [(x-11)/(x^3)] + 3x. However, for this problem, luckily, it doesn't matter.
The domain of the function includes all x-values that cause the function to have an output value. For rational functions such as F(x), recall that the denominator of the function cannot equal 0. Therefore, a domain value will not exist if it causes the denominator to equal 0.
Taking the denominator and setting it equal to 0, we get x3+3x = 0, and we now factor this:
x(x2+3) = 0
x = 0 or x2+3 = 0
x = 0 is a real value, whereas the second equation produces a pair of imaginary solutions, so we ignore them for the time being. This leaves us with x = 0 as the ONLY value of x that will cause the denominator to be a 0. Therefore, the domain of F(x) is ALL real numbers except for 0.
Domain of F(x): (- ∞, 0) U (0, ∞)
