For f(x) = ∛(x − h) + k, which statement is true about its domain and range?

• A. Domain and range are all real numbers
• B. Domain is x ≥ h; Range is y ≥ k
• C. Domain is all real numbers; Range is y all real numbers
• D. Domain is x ≤ h; Range is y ≤ k

Answer: (A)

The main idea is that cube root functions can take any real input and produce any real output, and shifting the graph doesn’t change that fundamental property.

For this function, x − h can be any real number as x runs through all real values, so the domain is all real numbers. The cube root of any real number is also a real number, so ∛(x − h) can produce every real value. Adding k shifts the outputs up or down, but you can still reach any real value by choosing the input accordingly: for any target y, pick x = h + (y − k)³, which gives f(x) = y. Therefore, the range is also all real numbers.