Question

consider the function $f(x) = -(x - 2)^3 + 8$. what is the domain of this function? (1 point)
○ $(-infty, 2)$
○ $(-infty, infty)$
○ $(2, infty)$
○ $(-infty, 2)$

Explanation:

Step1: Identify Function Type

The function \( f(x) = -(x - 2)^3 + 8 \) is a polynomial function (specifically a cubic polynomial).

Step2: Recall Domain of Polynomials

The domain of any polynomial function (including linear, quadratic, cubic, etc.) is all real numbers. This is because there are no restrictions (like division by zero, square roots of negative numbers, etc.) on the input \( x \) for polynomial functions—any real number can be substituted for \( x \) and the function will produce a real output.

Step3: Match with Options

The interval representing all real numbers is \( (-\infty, \infty) \).

Answer:

\( (-\infty, \infty) \) (the second option: \( (-\infty, \infty) \))