Short Answer 
The domain of a function defines the set of possible values for (x). Various cases show different domains, such as (y=sqrt[3]{x-2}) having a domain of all real numbers, while (y=sqrt{x-2}) only includes values (x geq 2). Confirming the inclusion of specific values is essential, as demonstrated by different conditions for various functions.
Step 1: Understanding Domain of Functions
The domain of a function refers to the complete set of possible values for x that the function can accept. This leads to a set of rules that help us determine if certain values can be included or excluded from the domain, especially when dealing with operations like square roots or fractions.
Step 2: Analyzing Different Cases
When investigating various functions, it is essential to look at their specific structures to find their domains. Consider these cases:
- Case 1: For the function (y=sqrt[3]{x-2}), the domain is all real numbers ((-‚Äöau, ‚Äöau)) since cube roots can take any value.
- Case 2: For (y=sqrt{x-2}), the radicand requires (x-2 geq 0), resulting in the domain ([2, ‚Äöau)).
- Case 3: The function (y=sqrt[3]{x+2}) also allows for all real numbers with a domain of ((-‚Äöau, ‚Äöau)).
- Case 4: Lastly, (y=sqrt{x+2}) needs the condition (x+2 geq 0), leading to the domain ([-2, ‚Äöau)).
Step 3: Confirming Inclusion of Values
Once the domains have been established, it’s crucial to confirm whether certain values such as zero are included. For instance, in Case 2, (x) must be greater than or equal to 2, making it undefined for (x=0). However, in Case 4, zero is included since (x) can equal -2 or any larger number, showing that the function operates for values starting from -2 upwards.