Algebra 2 Honors - Unit 5


5.1 - Rational Numbers as Exponents

For ALL EXPONENTS:

xmk=xmk=(xk)mx^{\frac{m}{k}}=\sqrt[k]{x^{m}}=\left(\sqrt[k]{x}\right)^{m}


For any rational number mn\frac{m}{n} any any positive real number x,

x−mn=1xmnx^{-\frac{m}{n}}=\frac{1}{x^{\frac{m}{n}}}


PRACTICE QUESTIONS ON NOTES WORKSHEET


5.2 - Properties of Exponents and Radicals

  1. Even roots:

    1. For xn\sqrt[n]{x} , n is the index and the x is the radicand

    2. If the index is even, the radicand is an even power, and the solution is an odd power, use ABSOLUTE VALUE

    3. Ex: with x6\sqrt{x^6} , the result. is x3x^3 . But the index is one and the radicand is even with 6, and the result has an odd power of 3. So we need to put absolute value

    4. So the actual answer is ∣x3∣\left|x^3\right|

  2. Odd roots:

    1. Since every real number has one odd root, if the index is odd, don’t use absolute value

    2. Ex: Since x2+43\sqrt[3]{x^2+4} has an odd root/index of 3, then we don’t need to do absolute value

  3. Domain for even roots: Since the radicand must non-negative for EVEN roots, the domain must keep the radicad non-negative

    1. So you can’t divide by 0 or take the square root of a negative number

    2. Ex: with f(x)=2−8xf\left(x\right)=\sqrt{2-8x} , 2-8x must be >= 0. So we solve and find the domain.

  4. Domain for odd roots: the domain for odd roots is all real numbers or D(−∞,∞)D\left(-\infty,\infty\right)


5.3 - Graphs of Radical Functions


Formulas:

f(x)=xf\left(x\right)=\sqrt{x} = y=abx−h+ky=a\sqrt{bx-h}+k


f(x)=x3f\left(x\right)=\sqrt[3]{x} = y=abx−h3+ky=a\sqrt[3]{bx-h}+k


PRACTICE AND REFERENCE NOTES WORKSHEET


5.5 - Radical Operations


  1. Theorem

    1. For any NONNEGATIVE numbers a and b and any natural number index k, ak⋅bk=abk\sqrt[k]{a}\cdot\sqrt[k]{b}=\sqrt[k]{ab}

    2. For any NONNEGATIVE a and b, any natural index k, b≠0b\ne0 , abk=akbk\sqrt[k]{\frac{a}{b}}=\frac{\sqrt[k]{a}}{\sqrt[k]{b}}

    3. IMPORTANT NOTE:

      1. For places when you’d usually have an absolute value for a variable, if that variable is both in and out of the radical, there’s no need for the absolute value since if it’s in the radical, then it must be greater than 0

      2. REFERENCE QUESTION 1 ON PAGE 2 OF NOTES

    4. Composition

      1. Learn how to find these domains with chat gpt

      2. Domains and ranges of equations like (f⋅g)(x)=f(g(x))\left(f\cdot g\right)\left(x\right)=f\left(g\left(x\right)\right)


5.4 - Radical Equations

  1. Extraneous Solutions - a solution which is true in a MODIFIED EQUATION, but FALSE in the original equation

    1. Ex: with x = 3 and x² = 9, x = -3, 3, but x = -3 is extraneous since it works when modified but not in the original


PRACTICE SOLVING PROBLEMS WITH MULTIPLE RADICALS

IN GENERAL REVIEW THE NOTES WORKSHEET


5.5 - Radical Operations


  1. Multiplying

    1. Radical expressions are multiplied like polynomials

  2. Division

    1. Division of radicals means rationalizing the denominator, or making it disappear

    2. PRACTICE

  3. Conjugates

    1. If the denominator has a sum or difference, you must multiply the top and bottom by the CONJUGATE of the BOTTOM ( the conjugate of a+b=a−b\sqrt{a}+\sqrt{b}=\sqrt{a}-\sqrt{b} )

    2. ALSO PRACTICE


IN GENERAL PRACTICE


5.6 Inverse Relations and Functions

  1. Inverse of Relations

    1. How to produce an INVERSE RELATION: Switch the x and y

    2. Ex: The inverse of (2,-3) is (-3,2)

  2. Graphing

    1. The graph of an inverse relation is a REFLECTION of the original relation across the line y = x

  3. One-To-One Functions

    1. In order for an inverse of a funtion to also be a function, the original function must be one-to-one

    2. That means that there’s only one value of x to one value of y

    3. The original function needs to pass the horizontal line test to be one-to-one

  4. Finding f−1(x)f^{-1}\left(x\right)

    1. Note any domain and range restrictions

      1. For the inverse, the original domain becomes the inverse range and the original range becomes the inverse domain

    2. Switch x and y

    3. Solve for y

    4. State the domain

    5. REFERENCE NOTES AND PRACTICE

  5. Testing for Inverse Functions

    1. PRACTICE, THiS IS A BIT UNCLEAR IN THE NOTES SO HAVE CHAT GPT EXPLAIN IT MORE

    2. Also:

      1. f(f−1(x))=xf\left(f^{-1}\left(x\right)\right)=x in the domain of f−1(x)f^{-1}\left(x\right) and vice versa


5.6b - Inverse Functions

JUST PRACTICE