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Try taking the square root of −8-8 on your calculator, and it throws an error. But ask for the cube root of −8-8, and it calmly hands you −2-2.
Why does that happen? Because multiplying three negative numbers yields a negative: (−2)×(−2)×(−2)=−8(-2) \times (-2) \times (-2) = -8.
The parent cube root function is defined algebraically as f(x)=x3f(x) = \sqrt[3]{x} or in exponential form as f(x)=x1/3f(x) = x^{1/3}.
General transformed cube root functions are typically written in the form f(x)=ab(x−h)3+kf(x) = a\sqrt[3]{b(x - h)} + k.
Because negatives are completely valid inputs and outputs, cube root functions behave totally differently from square roots. What does this mean for their domain and range?

Domain and Range

The radicand is the expression resting inside the radical symbol. For any cube root function f(x)=ab(x−h)3+kf(x) = a\sqrt[3]{b(x - h)} + k, the radicand can be positive, negative, or zero.
Because every real number has exactly one real cube root, the domain is all real numbers, or (−∞,∞)(-\infty, \infty). Likewise, outputs stretch to positive and negative infinity, so the range is also always (−∞,∞)(-\infty, \infty).
Square roots have a hard endpoint where the graph begins. What takes that place in a cube root function?

Symmetry and the Inflection Point

The parent function f(x)=x3f(x) = \sqrt[3]{x} has origin symmetry, meaning it is an odd function where f(−x)=−f(x)f(-x) = -f(x). Rotating the graph 180∘180^\circ around the origin (0,0)(0, 0) lands the curve right back onto itself.