AP Precalculus: Radical Expressions & Functions Formulas
1. Simplifying Radical Expressions with Variables
- \( \sqrt{a^2} = |a| \)
- \( \sqrt[n]{a^n} = |a| \) (if \(n\) even), \( \sqrt[n]{a^n} = a \) (if \(n\) odd)
- Product: \( \sqrt{ab} = \sqrt{a} \sqrt{b} \)
- Quotient: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)
- Powers: \( \sqrt[n]{a^m} = a^{\frac{m}{n}} \)
2. Nth Roots
- \( \sqrt[n]{a} = a^{1/n} \)
- \( n \) even: \( a \ge 0 \); \( n \) odd: all real \( a \)
3. Domain and Range of Radical Functions
- \( f(x) = \sqrt{x-h} + k \):
Domain: \( x \ge h \); Range: \( y \ge k \) - \( f(x) = \sqrt[n]{x} \) (odd \(n\)): domain/range is \( \mathbb{R} \) (all real numbers)
- Set inside ≥ 0 for even roots: \( \sqrt[n]{g(x)} \implies g(x) \ge 0 \)
4. Graphing Square Root Functions
- Parent: \( f(x) = \sqrt{x} \) (starts at (0,0), rises rightward)
- General: \( f(x) = a\sqrt{x-h} + k \)
- Right by \(h\), up by \(k\), reflected if \(a<0\)
- Domain: \(x \ge h\); Range \( y \ge k \) if \(a > 0\)
5. Solving Radical Equations
- Isolate radical, then raise both sides to power \(n\) to eliminate: \( \sqrt[n]{f(x)} = a \implies f(x) = a^n \)
- Solve for \(x\), then check for extraneous solutions (plug answers back)