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Cube Root Function — Definition, Formula & Examples

The cube root function is the function f(x)=x3f(x) = \sqrt[3]{x}. It takes any real number and returns the number that, multiplied by itself three times, gives back the input. For example, f(8)=2f(8) = 2 and f(−8)=−2f(-8) = -2.

The cube root function f:R→Rf: \mathbb{R} \to \mathbb{R} is defined by f(x)=x3f(x) = \sqrt[3]{x}, where y=x3y = \sqrt[3]{x} is the unique real number satisfying y3=xy^3 = x. It is the inverse of g(x)=x3g(x) = x^3. Its domain and range are both (−∞,∞)(-\infty, \infty). It is odd, since f(−x)=−f(x)f(-x) = -f(x), increasing everywhere, and continuous everywhere. Its graph passes through (0,0)(0,0) with a vertical tangent and an inflection point there.

Key Formula

f(x)=x3,(x3)3=x for every real xf(x) = \sqrt[3]{x}, \qquad \left(\sqrt[3]{x}\right)^3 = x \text{ for every real } x
Where:
  • xx = the input, which can be any real number, including negatives and zero
  • f(x)f(x) = the output, the real cube root of x

Worked Example

Problem: Let g(x)=x−23+1g(x) = \sqrt[3]{x - 2} + 1. Find g(10)g(10) and g(−6)g(-6). Then find the value of xx for which g(x)=4g(x) = 4.
Step 1: Substitute x=10x = 10. The inside becomes 88, and 83=2\sqrt[3]{8} = 2 because 23=82^3 = 8.
g(10)=10−23+1=83+1=2+1=3g(10) = \sqrt[3]{10 - 2} + 1 = \sqrt[3]{8} + 1 = 2 + 1 = 3
Step 2: Substitute x=−6x = -6. The inside is negative, which is allowed. −83=−2\sqrt[3]{-8} = -2 because (−2)3=−8(-2)^3 = -8.
g(−6)=−6−23+1=−83+1=−2+1=−1g(-6) = \sqrt[3]{-6 - 2} + 1 = \sqrt[3]{-8} + 1 = -2 + 1 = -1
Step 3: Set g(x)=4g(x) = 4, then isolate the cube root and cube both sides.
x−23+1=4  ⇒  x−23=3  ⇒  x−2=27  ⇒  x=29\sqrt[3]{x - 2} + 1 = 4 \;\Rightarrow\; \sqrt[3]{x - 2} = 3 \;\Rightarrow\; x - 2 = 27 \;\Rightarrow\; x = 29
Answer: g(10)=3g(10) = 3, g(−6)=−1g(-6) = -1, and g(x)=4g(x) = 4 when x=29x = 29.

Why It Matters

In Algebra 2 and Precalculus, y=x3y = \sqrt[3]{x} is one of the parent functions you shift, stretch, and reflect. In this example, g(x)=x−23+1g(x) = \sqrt[3]{x-2}+1 is the parent graph moved 2 right and 1 up. It also answers real questions such as finding the side length of a cube from its volume: a 64 cm³ box has sides of 643=4\sqrt[3]{64} = 4 cm.

Common Mistakes

Mistake: Treating the cube root function like the square root function and restricting its domain to x≥0x \ge 0.
Correction: A negative number times itself three times stays negative, so negative inputs have real cube roots. For example, −273=−3\sqrt[3]{-27} = -3. The domain is all real numbers, and the graph extends into the third quadrant.

Related Terms