Square Root, Cube Root, Cubic & Absolute Value Functions
Four parent functions, four shapes, four sets of rules. Domain restrictions for square roots, the symmetry of cube roots, the V of absolute value, and how transformations apply uniformly to all of them.
Same template, different shapes
Each of these parent functions follows the same transformation rules from the function-transformations lesson. What changes is the shape of the parent and the domain restrictions you have to watch for.
Square root: domain restricted
For f(x) = √(x − 4), the inside (x − 4) must be ≥ 0, so x ≥ 4. The graph starts at (4, 0) and curves to the right.
Solving radical equations
Squaring can introduce solutions that don't satisfy the original equation. Always plug back to verify.
Cube root: defined for all real numbers
Unlike square root, cube root accepts any input — even negatives. ∛(−8) = −2 because (−2)³ = −8.
Square root: domain x ≥ 0. Cube root: domain all reals. The CBE often tests this distinction directly.
Absolute value: distance from zero
|a| = a if a ≥ 0, and |a| = −a if a < 0. The graph of y = |x| is a V-shape with vertex at the origin.
Solving absolute value equations
If |something| = k (where k > 0), the inside can equal +k or −k. Two equations, two solutions.
3-second recap
- √: domain x ≥ 0. Solve by squaring, always check.
- ∛: domain all reals. Solve by cubing.
- x³: S-shape; one real root for any equation x³ = k.
- |x| = k: two equations (inside = ±k); zero solutions if k < 0.