### Instruction: Find the domain of the function $z(x) = \sqrt[3]{x - 1} + \sqrt[3]{8 - x}.$.### Response: To find the domain of a function, I need to identify the values of x that make the function undefined or invalid.
One thing that can make a function undefined is a zero denominator, but I don't see any fractions here, so that's not an issue.
Another thing that can make a function invalid is taking the square root of a negative number, but I notice that this function has cube roots, not square roots.
Cube roots can handle negative numbers, so there's no problem there either.
So, it seems that this function is well-defined and valid for any value of x.
Therefore, the domain of z(x) is all real numbers.