The given linear transformation dx maps a function f(x) to its derivative, dx(f) = 4x^2. To find the preimage, we need to determine the original function f(x) that satisfies this derivative.
By integrating the derivative with respect to x, we can find the antiderivative F(x) of 4x^2. The antiderivative is obtained by reversing the process of differentiation.
The antiderivative of 4x^2 is (4/3)x^3, where (4/3) is the coefficient of the term and x^3 is the term raised to the power one higher than the exponent in the derivative. The constant of integration C is added to account for the family of functions that have the same derivative.
Therefore, the preimage of the function dx(f) = 4x^2 is F(x) = (4/3)x^3 + C, where C represents any constant value.