📌 What is a Radical Function?

A radical function is a function that contains a variable inside a radical (root) symbol:

• Square Root Function: \( f(x) = \sqrt{x} \)
• Cube Root Function: \( f(x) = \sqrt[3]{x} \)
• General Form: \( f(x) = a\sqrt[n]{x-h} + k \)

Square Root Function: \( f(x) = \sqrt{x} \)

Cube Root Function: \( f(x) = \sqrt[3]{x} \)

Finding Domain of Radical Functions:

For square root functions (even index), set the radicand \( \geq 0 \):

If \( f(x) = \sqrt{g(x)} \), then \( g(x) \geq 0 \)

For cube root functions (odd index), all real numbers are allowed:

If \( f(x) = \sqrt[3]{g(x)} \), then domain is all real numbers

📝 Examples - Finding Domain:

Example 1: Find domain of \( f(x) = \sqrt{x - 3} \)

Set radicand \( \geq 0 \): \( x - 3 \geq 0 \)
Solve: \( x \geq 3 \)
Domain: \( [3, \infty) \)

Example 2: Find domain of \( f(x) = \sqrt{5 - 2x} \)

Set radicand \( \geq 0 \): \( 5 - 2x \geq 0 \)
Solve: \( -2x \geq -5 \)
\( x \leq \frac{5}{2} \)
Domain: \( (-\infty, \frac{5}{2}] \)

Example 3: Find domain of \( f(x) = \sqrt[3]{x + 2} \)

Cube root accepts all real numbers
Domain: \( (-\infty, \infty) \)

Standard Form of Square Root Function:

\( f(x) = a\sqrt{x - h} + k \)

Key Points for Graphing Square Root Functions:

1. Identify the starting point: \( (h, k) \)
2. Create a table of values: Use perfect squares for easy calculation
3. Apply transformations: stretch/compress, then shift
4. Plot points and connect with smooth curve
5. Determine domain and range from the graph

📝 Example - Graphing Square Root:

Graph \( f(x) = 2\sqrt{x - 1} + 3 \)

Step 1: Identify transformations:

• \( a = 2 \) → vertical stretch by factor of 2
• \( h = 1 \) → shift right 1 unit
• \( k = 3 \) → shift up 3 units

Step 2: Starting point: \( (1, 3) \)

Step 3: Key points:

\( x = 1 \): \( f(1) = 2\sqrt{0} + 3 = 3 \) → \( (1, 3) \)
\( x = 2 \): \( f(2) = 2\sqrt{1} + 3 = 5 \) → \( (2, 5) \)
\( x = 5 \): \( f(5) = 2\sqrt{4} + 3 = 7 \) → \( (5, 7) \)
\( x = 10 \): \( f(10) = 2\sqrt{9} + 3 = 9 \) → \( (10, 9) \)

Domain: \( [1, \infty) \), Range: \( [3, \infty) \)

Standard Form of Cube Root Function:

\( f(x) = a\sqrt[3]{x - h} + k \)

Key Points for Cube Root Parent Function:

The parent function \( f(x) = \sqrt[3]{x} \) passes through these key points:

📝 Example - Graphing Cube Root:

Graph \( f(x) = -\sqrt[3]{x + 2} + 1 \)

Step 1: Identify transformations:

• \( a = -1 \) → reflection over x-axis
• \( h = -2 \) → shift left 2 units
• \( k = 1 \) → shift up 1 unit

Step 2: Center point: \( (-2, 1) \)

Step 3: Key points:

\( x = -10 \): \( f(-10) = -\sqrt[3]{-8} + 1 = -(-2) + 1 = 3 \) → \( (-10, 3) \)
\( x = -3 \): \( f(-3) = -\sqrt[3]{-1} + 1 = -(-1) + 1 = 2 \) → \( (-3, 2) \)
\( x = -2 \): \( f(-2) = -\sqrt[3]{0} + 1 = 1 \) → \( (-2, 1) \)
\( x = -1 \): \( f(-1) = -\sqrt[3]{1} + 1 = 0 \) → \( (-1, 0) \)
\( x = 6 \): \( f(6) = -\sqrt[3]{8} + 1 = -2 + 1 = -1 \) → \( (6, -1) \)

Domain: \( (-\infty, \infty) \), Range: \( (-\infty, \infty) \)

What is a Radical Equation?

A radical equation is an equation in which the variable appears under a radical symbol.

Steps to Solve Radical Equations:

1. Isolate the radical on one side of the equation
2. Raise both sides to the power that matches the index of the radical
   • Square both sides for square roots
   • Cube both sides for cube roots
3. Solve the resulting equation
4. Check ALL solutions in the original equation (very important!)
5. Eliminate extraneous solutions (solutions that don't work)

⚠️ Extraneous Solutions:

An extraneous solution is a solution obtained algebraically that does NOT satisfy the original equation.

📝 Example 1 - Simple Radical Equation:

Solve: \( \sqrt{x + 5} = 7 \)

Step 1: Radical is already isolated

Step 2: Square both sides

\( (\sqrt{x + 5})^2 = 7^2 \)
\( x + 5 = 49 \)

Step 3: Solve for \(x\)

\( x = 44 \)

Step 4: Check in original equation

\( \sqrt{44 + 5} = \sqrt{49} = 7 \) ✓

Solution: \( x = 44 \)

📝 Example 2 - With Extraneous Solution:

Solve: \( \sqrt{2 - x} = x \)

Step 1: Radical is isolated

Step 2: Square both sides

\( (\sqrt{2 - x})^2 = x^2 \)
\( 2 - x = x^2 \)

Step 3: Rearrange and solve

\( x^2 + x - 2 = 0 \)
\( (x + 2)(x - 1) = 0 \)
\( x = -2 \) or \( x = 1 \)

Step 4: Check both solutions

Check \( x = -2 \): \( \sqrt{2-(-2)} = \sqrt{4} = 2 \neq -2 \) ✗ (extraneous)
Check \( x = 1 \): \( \sqrt{2-1} = \sqrt{1} = 1 = 1 \) ✓

Solution: \( x = 1 \) (reject \( x = -2 \) as extraneous)

📝 Example 3 - Isolate First:

Solve: \( \sqrt{x + 2} + 4 = x \)

Step 1: Isolate the radical

\( \sqrt{x + 2} = x - 4 \)

Step 2: Square both sides

\( (\sqrt{x + 2})^2 = (x - 4)^2 \)
\( x + 2 = x^2 - 8x + 16 \)

Step 3: Rearrange and solve

\( 0 = x^2 - 9x + 14 \)
\( 0 = (x - 7)(x - 2) \)
\( x = 7 \) or \( x = 2 \)

Step 4: Check both solutions

Check \( x = 7 \): \( \sqrt{7+2} + 4 = 3 + 4 = 7 = 7 \) ✓
Check \( x = 2 \): \( \sqrt{2+2} + 4 = 2 + 4 = 6 \neq 2 \) ✗ (extraneous)

Solution: \( x = 7 \)

📝 Example 4 - Cube Root Equation:

Solve: \( \sqrt[3]{2x - 5} = 3 \)

Step 1: Radical is isolated

Step 2: Cube both sides

\( (\sqrt[3]{2x - 5})^3 = 3^3 \)
\( 2x - 5 = 27 \)

Step 3: Solve for \(x\)

\( 2x = 32 \)
\( x = 16 \)

Step 4: Check

\( \sqrt[3]{2(16) - 5} = \sqrt[3]{27} = 3 \) ✓

Solution: \( x = 16 \)

Note: Cube root equations rarely have extraneous solutions (but still check!)

⚡ Quick Summary

• Square root function domain: Set radicand \( \geq 0 \)
• Cube root function domain: All real numbers
• Standard form: \( f(x) = a\sqrt[n]{x-h} + k \)
• \(a\): vertical stretch/compression and reflection
• \(h\): horizontal shift (opposite sign)
• \(k\): vertical shift
• To solve radical equations: Isolate, raise to power, solve, CHECK!
• Extraneous solutions: Algebraic solutions that don't work in original
• Always verify solutions by substituting into original equation
• Square root graphs start at a point and curve upward
• Cube root graphs extend in both directions through center point

📚 Key Formulas Reference