f(x) = (2x)/ [(x^2 + x + 1)]^(1/2)

Question

How would I find the domain of f?

Answer 1

The expression to worry about is the denominator √(x^2 + x + 1)

First of all the denominator can not be zero. Solving x^2 + x + 1 = 0 gives me two complex roots.
So it can never be zero.

The second consideration is to make sure the inside of the square root is not negative. By finding complex roots above guaranteed that the inside cannot be negative.
Another way would be to consider the parabola y = x^2 + x + 1, which opens up and has its vertex above the x-axis
So the value inside the √ sign cannot be either zero or some negative.

So the domain for your function is the set of real numbers,
anything goes.