In Exercises 19–32, find the (a) domain and (b) range.
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𝔂 = -1 + ∛ 2 - x

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the given function 𝔂 = -1 + ∛(2 - x), we need to determine the values of x that do not lead to undefined expressions. Since the cube root function is defined for all real numbers, the domain is all real numbers where 2 - x is a real number, which translates to x being less than or equal to 2.

The range of a function is the set of all possible output values (y-values) that the function can produce. In the case of 𝔂 = -1 + ∛(2 - x), as x approaches negative infinity, the cube root term will also approach negative infinity, and as x approaches 2, the output approaches -1. Therefore, the range of this function is all real numbers greater than or equal to -1.

The cube root function, denoted as ∛x, is a function that returns the number which, when cubed, gives the input x. This function is defined for all real numbers, meaning it can take any real number as input and will produce a real number as output. Understanding the properties of the cube root function is essential for analyzing the behavior of the given function, particularly in determining its domain and range.