Final answer:
The function with simple poles at 0 and -2, a double pole at -3, and zeros at -1 and infinity can be represented as f(x) = K*(x + 1)/((x^2)(x + 2)(x + 3)^2), where K is an unspecified constant.
Step-by-step explanation:
The question requires us to find a function given its poles and zeros. For this case, we have simple poles at 0 and -2, a double pole at -3, and zeros at -1 and infinity. A function with these characteristics could be constructed using the product of terms that specify its zeros and poles.
Formally, the function can be expressed as f(x) = K*(x + 1)/((x^2)(x + 2)(x + 3)^2), where K is a constant that can be determined based on additional conditions or normalization requirements not specified here. The numerator represents the zero at -1, while the denominator includes terms for each pole. Notably, the zero at infinity does not manifest as a term in the function, as it implies the function approaches zero as x approaches infinity.