Consider f(x) = ∛(x − 4) + 2. Which statement is true about its domain and range?

Consider f(x) = ∛(x − 4) + 2. Which statement is true about its domain and range?

• A. Domain is x ≥ 4; Range is y ≥ 2
• B. Domain and range all real numbers
• C. Domain is x ≤ 4; Range is y ≤ 2
• D. Domain is all real numbers; Range is y ≤ 2

Answer: (B)

The key idea is that a cube root has no input restriction. ∛(x−4) is defined for every real x, since x−4 can be any real number. Adding 2 just shifts all outputs upward by 2, so the set of possible y-values remains all real numbers as well. Therefore, both the domain and the range are all real numbers. The other descriptions would impose bounds that don’t actually occur—you can plug in any real x and get a real output, including values far less than 2.

The key idea is that a cube root has no input restriction. ∛(x−4) is defined for every real x, since x−4 can be any real number. Adding 2 just shifts all outputs upward by 2, so the set of possible y-values remains all real numbers as well. Therefore, both the domain and the range are all real numbers. The other descriptions would impose bounds that don’t actually occur—you can plug in any real x and get a real output, including values far less than 2.