Algebra 2
Other functions (advanced)
Cube and cube root functions
20 practice questions
0 video lessons
Theory + worked examples
Cube and Cube Root Functions
Texas Algebra II (TEKS) • Standard 2A.6(A), 2A.6(B) • Other Functions
Cube and Cube Root Functions is a topic in Other Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.6(A), 2A.6(B), which requires students to analyze the effect of parameter changes on cubic and cube root functions and solve cube root equations.
The cube \(x^3\) and cube root \(\sqrt[3]{x}\) are inverses, both defined for all reals, odd, and passing through the origin.
Theory
The cube and cube root functions are inverses:
- \(x^3\): domain and range all reals; odd; through the origin.
- \(\sqrt[3]{x}\): undoes the cube; also all reals.
- Both transform as \((x-h)^3+k\) and \(\sqrt[3]{x-h}+k\).
Unlike square roots, cube roots accept negative numbers.
\(x^3\) and \(\sqrt[3]{x}\) are inverses through the origin.
Cube and cube root features.
Inverse operations:
\[x^3=a\ \Rightarrow\ x=\sqrt[3]{a},\qquad \sqrt[3]{x}=a\ \Rightarrow\ x=a^3\]
Cube roots of negatives are real: \(\sqrt[3]{-8}=-2\).
How to solve
- To undo a cube, take the cube root.
- To undo a cube root, cube both sides.
- Keep the sign — odd powers preserve it.
- For transforms, read \((h,k)\).
Example 1 — Solve a cubic
Solve \(x^3=27\).
Solution
Take the cube root.
| \(x\) | \(=\) | \(\sqrt[3]{27}=3\) |
Example 2 — Solve a cube root
Solve \(\sqrt[3]{x}=2\).
Solution
Cube both sides.
| \(x\) | \(=\) | \(2^3=8\) |
Example 3 — Negative cube
Solve \(x^3=-8\).
Solution
An odd power keeps the sign.
| \(x\) | \(=\) | \(\sqrt[3]{-8}=-2\) |
Example 4 — Transformed cubic
Describe \(f(x)=(x-1)^3+2\).
Solution
Shift the parent right \(1\), up \(2\).
| \(\text{center}\) | \(=\) | \((1,2)\) |
Common pitfalls
Cube roots of negatives are real, unlike square roots.
An odd power keeps the sign of the input.
No extraneous solutions from cubing — it is reversible.
Frequently asked questions
What is the domain of \(x^3\)?
All real numbers.
Can you take the cube root of a negative?
Yes — \(\sqrt[3]{-8}=-2\).
How do you solve \(x^3=27\)?
Take the cube root: \(x=3\).
Are the cube and cube root inverses?
Yes — each undoes the other.
More in Other functions (advanced)