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

Verified step by step guidance

1. Step 1: Identify the function type. The given function is 𝑦 = -1 + ∛(2 - x), which involves a cube root. Cube root functions are defined for all real numbers, so there are no restrictions on the domain.
2. Step 2: Determine the domain. Since the cube root function is defined for all real numbers, the domain of 𝑦 = -1 + ∛(2 - x) is all real numbers, or (-∞, ∞).
3. Step 3: Analyze the range of the function. The cube root function, ∛(x), can take any real number as its output, ranging from -∞ to ∞.
4. Step 4: Consider the transformations applied to the cube root function. The function 𝑦 = -1 + ∛(2 - x) involves a vertical shift downward by 1 unit. This does not affect the range of the cube root function, which remains all real numbers.
5. Step 5: Conclude the range. Since the cube root function can output any real number and the vertical shift does not restrict this, the range of the function is also all real numbers, or (-∞, ∞).

Key Concepts

Domain

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.

Range

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.

Cube Root Function

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.