Radicals

Radicals

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Radicals by Mind Map: Radicals

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)."