## 1. Vocabulary

### 1.1. Radical: A radical symbol with an index and radicand.

### 1.2. Radical Symbol: Also known as a root sign, the symbol representing a radical, which looks like a check mark.

### 1.3. Index: The degree of the radical, index root of x = x^(1/index).

### 1.4. Radicand: The number inside of the radical symbol.

### 1.5. Coefficient: The multiplier of the radical (or a variable in general).

### 1.6. Absolute Value: When a number is in absolute value signs, it is positive.

1.6.1. |-3| = 3

### 1.7. Numerator: The numerator is the upper number in a fraction.

### 1.8. Denominator: The numerator is the lower number in a fraction.

### 1.9. Extraneous Solution: A solution which can be obtained through algebraic steps, but does not satisfy the radical equation.

### 1.10. Conjugate: Two terms with a switched sign in the middle.

## 2. Radical Expressions

### 2.1. Radicals

2.1.1. A radical represents a number that, when raised to the power of the index, equals the radicand.

2.1.1.1. √(9) = 3

### 2.2. Adding & Subtracting Radicals

2.2.1. Radical Equations are like variables when adding and subtracting, different ones cannot be combined.

2.2.1.1. 2√(2) + 3√(7) - √(2) = √(2) + 3√(7)

### 2.3. Multiplying & Dividing Radicals

2.3.1. Radical equations, like exponents, can be combined when multiplied or divided.

2.3.1.1. √(x) * √(y) = √(x * y)

2.3.1.2. √(x) / √(y) = √(x / y)

## 3. Simplifying/Solving Radical Expressions

### 3.1. Rationalizing the Denominator

3.1.1. If the denominator of a fraction contains a radical, you can multiply the numerator and denominator by the conjugate of the denominator, if it has two terms.

3.1.1.1. (5) / (2 - √3)

3.1.1.1.1. ((5) * (2 + √3)) / ((2 - √3) * (2 + √3))

### 3.2. Distributing Radicals

3.2.1. In radicals, like in exponents, the radicand can be divided into muliple parts, and then distrubuted between radicals with the same index.

3.2.1.1. √(50)

3.2.1.1.1. √(25 * 2)

### 3.3. Absolute Value Signs

3.3.1. When simplifying variable expression from radicals, you must use absolut value signs around the expression if:

3.3.1.1. The index of the radical was even.

3.3.1.2. ...and...

3.3.1.3. The exponent of the derived expression is odd.

## 4. ...and some other stuff.

### 4.1. Composition of Functions

4.1.1. Composition of Functions is where one function's output acts as the input for another function.

4.1.1.1. (f o g)(x) = f(g(x))

4.1.1.1.1. For example, if:

4.1.1.1.2. Then:

4.1.1.1.3. So: (f o g)(x) = -3(x^2) + 7

4.1.2. Joke:

4.1.2.1. If: "In tragedy, it's hard to find a good resolution; it's not black and white: it's a big fog of gray."

4.1.2.2. "fog of gray" --> "(f o g) of gray" --> "(f o g)(gray)" --> "f(g(gray))"

4.1.2.3. So: "In tragedy, it's hard to find a good resolution; it's not black and white: it's a big f(g(gray))."

### 4.2. Inverse Functions

4.2.1. The inverse of a function is the function flipped over the line, x = y.

4.2.2. The inverse of a function takes the output of the function, and turns it into the input.

4.2.2.1. The inverse of a function can be found by solving for x, and then switching x and y.

4.2.2.1.1. For example, if:

4.2.2.1.2. Then: y = 2x - 18 --> y + 18 = 2x --> y/2 + 9 = x

4.2.2.1.3. So: (f^-1)(x) = x/2 + 9

4.2.2.2. If: f(10) = 2

4.2.2.3. Then: (f^-1)(2) = 10

4.2.2.4. So: f(p) = q --> (f^-1)(q) = p

4.2.3. The notation for the inverse of f(x) is (f^-1)(x).

## 5. Graph Radical Equations

### 5.1. From the base equation, y = √x:

5.1.1. You can make the equation y = m√((x / n) - h) + k.

5.1.1.1. m is the vertical stretch.

5.1.1.2. n is the horizontal stretch.

5.1.1.3. h is the positive horizontal translation.

5.1.1.4. k is the positive vertical translation.

### 5.2. Domain:

5.2.1. If the index is odd, then the domain is "all real numbers."

5.2.2. If the index is even, then the domain is "x is greater than or equal to h * sign(n)."

### 5.3. Range:

5.3.1. If the index is odd, then the range is "all real numbers."

5.3.2. If the index is even, then the range is "y is greater than or equal to k * sign(m)."