___The domain of f(x) = ³√x−4 is [4, ∞).

Verified step by step guidance

1. Identify the function: \( f(x) = \sqrt[3]{x-4} \).
2. Understand that the cube root function \( \sqrt[3]{x} \) is defined for all real numbers.
3. Recognize that the expression inside the cube root, \( x-4 \), must be defined for all real numbers.
4. Since there are no restrictions on \( x \) for the cube root, \( x-4 \) can be any real number.
5. Conclude that the domain of \( f(x) = \sqrt[3]{x-4} \) is all real numbers, which is \((-\infty, \infty)\).

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = ³√x−4, the domain is determined by the values of x that do not lead to undefined expressions. In this case, since the cube root function is defined for all real numbers, the domain starts from 4 and extends to positive infinity.

Cube Root Function

The cube root function, denoted as ³√x, is the inverse operation of cubing a number. It takes a real number x and returns a value y such that y³ = x. Unlike square roots, cube roots can accept negative values, which means they are defined for all real numbers. This property is crucial in determining the domain of the function f(x) = ³√x−4.

Vertical Shift

A vertical shift occurs when a constant is added or subtracted from a function, affecting its output values without altering its input values. In the function f(x) = ³√x−4, the '-4' indicates a downward shift of the cube root function by 4 units. This shift does not impact the domain but changes the range of the function, which is important for understanding its overall behavior.