Algebra 2 · Unit 5 · Topic 5.5
Square-Root and Cube-Root Functions
The real square-root function has domain [0, ∞) and nonnegative outputs.
Definition and explanation
The real square-root function has domain [0, ∞) and nonnegative outputs. The real cube-root function has all real inputs and outputs. Shifts and reflections change the displayed domain and range accordingly.
Worked example
For f(x) = 2√(x − 3) − 4, the domain is x ≥ 3 and the range is y ≥ −4. Its starting point is (3, −4); at x = 7 it gives y = 0. For g(x) = ∛(x + 2), there is no real-input restriction.
Common mistake
A square-root graph and a cube-root graph have different domains; they are not interchangeable.
Key takeaway
The real square-root function has domain [0, ∞) and nonnegative outputs.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.