Domain is defined as all real values of x (input) for which a function doesn't fail
Function will fail if the output becomes an Imaginary number or is Un-defined

Let's have a look at the option choices:

A. \(f(x)=\frac{4}{x+4}\)
\(x ≠ -4\), else f(x) is undefined
So, x∈R except -4

B.\( f(x)=\frac{4}{x^2+1}\)
\(x^2 + 1\) can never be zero i.e. f(x) will always have some output value
So, x∈R

C. \(f(x)=\sqrt{4x^3}\)
Here, x cannot be a -ve number, else f(x) is an imaginary number
So, x∈R, where x ≥ 0

D. \(f(x)=\sqrt{(-4x)^2} = -4x\)
f(x) can never be an imaginary number i.e. any value of x will always yield a valid value of f(x)
So, x∈R

E. \(f(x)=\sqrt[3]{4x} = (4x)^{\frac{1}{3}}\)
f(x) can never be an imaginary number i.e. any value of x will always yield a valid value of f(x)
So, x∈R

Hence, option B, D and E