In order to find the domain of the given function, we need find for which x-values the function is defined. The issue would be if the denominator equals 0, which would make the function undefined.
So, we have to find the solutions of equation 3x^3 + 24 = 0.
To solve for x, we need to get 3x³ on its own. We do this by subtracting 24 from both sides of the equation to isolate 3x^3:
3x^3 = -24
Then we divide both sides of the equation by 3 to isolate x^3:
x^3 = -24/3
We simplify the right-hand side to get:
x^3 = -8
Now, to solve for x, we could take the cube root of both sides to get
x = -2
So the function is not defined when x = -2.
Now let's figure out the domain. The domain of the function will be all real numbers except for x = -2. In interval notation, we can represent this as (-∞, -2) U (-2, ∞). "U" denotes the union of two sets, meaning that the domain includes all numbers from negative infinity to -2, and from -2 to positive infinity, but does not include -2 itself.
So, the domain of the function is all real x, except x=-2.