Final answer:
The domain and range of the function f(x) =(x-2)³ +1 are both all real numbers, since there are no restrictions on the input and cubic functions can produce any real number output.
Step-by-step explanation:
The domain of a function refers to the set of all possible input values for which the function is defined, whereas the range of a function refers to the set of possible output values. Considering the function f(x) =(x-2)³ +1, there are no restrictions on the input value x, since for any real number input, the function produces a real number output. Therefore, the domain of f(x) is all real numbers, or ∞ < x < ∞. As for the range, the function is a cubic function shifted upward by 1. Cubic functions have the property that they can take on any real value depending on the input, so the range is also all real numbers.
Thus, the domain of f(x) is: ∞ < x < ∞ (all real numbers)
And the range of f(x) is: ∞ < f(x) < ∞ (all real numbers)