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Learn · Algebra 2

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.

9 minTEKS 6A,6B,6F,6LAlgebra 2

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: y = √x — starts at origin, opens right (domain x ≥ 0)start (0, 0)(4, 2)(9, 3)domain: [0, ∞)range: [0, ∞)

Square root: domain restricted

f(x) = √xDomain: x ≥ 0 (you can't take an even root of a negative)Range: y ≥ 0
Watch the domain

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.

Find the domain
What is the domain of f(x) = √(x − 4)?

Solving radical equations

√(x + 5) = 3(√(x + 5))² = 3² (square both sides)x + 5 = 9x = 4Always check by substituting back: √(4+5) = √9 = 3 ✓.
Extraneous solutions

Squaring can introduce solutions that don't satisfy the original equation. Always plug back to verify.

Solve a square root equation
Solve the square root equation √(x + 5) = 3 for x.
Cubic: y = x³ — passes through origin, S-shape, odd symmetryinflection (0, 0)(1, 1)(−1, −1)rapid growth →rapid drop ←domain (−∞, ∞); range (−∞, ∞); f(−x) = −f(x) → odd function

Cube root: defined for all real numbers

Unlike square root, cube root accepts any input — even negatives. ∛(−8) = −2 because (−2)³ = −8.

Key difference

Square root: domain x ≥ 0. Cube root: domain all reals. The CBE often tests this distinction directly.

Compare domains
Which is true for f(x) = ∛x (cube root)?
Absolute value: y = |x| — sharp V at vertex, always non-negativevertexleft branch: y = −xright branch: y = xdomain (−∞, ∞); range [0, ∞); f(−x) = f(x) → even function

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.

|2x − 5| = 72x − 5 = 7 or 2x − 5 = −72x = 12 → x = 62x = −2 → x = −1x = 6 or x = −1
Solve an absolute value equation
Solve the absolute value equation |2x − 5| = 7 for all values of x.

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.

Check yourself

Quick check #1
What is the domain of f(x) = √(x − 5)?
Quick check #2
Solve |2x − 3| = 7.