Chapter 8: Inverses and Radicals

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Wiki Summary Assignments:

8-1: Justin F. and Austin L.
8-2: Crystalene E. and Dakota E.
8-3: Elissa M. and Dale N.
8-4: Marissa B. and Brittany K.
8-5: Nate F. and Sarah P.
8-6: Shannon M. and Max S.
8-7: Kaitlynn R. and Caiti T.
8-8: Ricky S. and Bonus

Section 8-1: Composition of Functions

Notes: aanotes8-1.pdf

Justin F. - Lesson 8-1 has to do with the composition of functions. Composite g of f - The function that maps x onto g(f(x)), whose domain is the set of all values in the domain of f, then also in g. g of (x) = g(f(x)) → g of f of x; Do f first, then g example: f(x) = 2x-1, g(x) = 1/x f(g(5)) = f(1/5) =2(1/5)-1 =2/5 -1 =-3/5

Austin L.- Lesson 8-1 deals with the composite of functions. The composite of two functions is a function. Composites can be written in 2 different ways. Mapping notation and euler's notation; we already know how these are written from previous lessons. Knowing the order of the function is vital to the understanding of composites of a function. g(f(x)), first you would solve for f(x) then you would use that equation and then put in g.
Composite g of f- The function that maps x onto g(f(x)), whose domain is the set of all values in the domain of f, then also in g.

Section 8-2: Inverses of Relations

Notes: aanotes8-2.pdf

Section 8-3: Properties of Inverse Functions

Notes: aanotes8-3.pdf

Dale N - In this lesson we learned about the properties of inverse functions. We learned the Inverse functions theorum where two functions f and g are inverse functions, if and only if: (1) For all x in the domain of f, g º f(x)=x and (2) For all x in the domain of g,f º g(x)=x. We also learned that the proof has two parts. The "only if" and the "if" part. When we have a power function, we use the Power function Inverse Theorum where if f(x)=x^n and g(x)=x^1/n and the domains of f and g are the set of nonnegative real numbers, then f and g are inverse functions.

In 8-3 we learned that to find an inverse you have to switch x and y. An inverse is something
that undoes something that was already done. Like taking three steps forward and then taking three steps back.
Inverse Function Theorem
1. For all x in the domain of F and g(f(x))=x
2. For all y in the domain of g and f(g(x))=x
No matter what these steps will take you back to x.
Inverse Function Notation:
For a function f(x), the inverse is f-1 (x)
Power Function Inverse Theorem:
If f(x)=x^n and g(x)=x^1/n and the domains of f and g are the set of non negative
real numbers then f and g are inverse functions.
Elissa M.

Section 8-4: Radical Notation for nth Roots

Notes: aanotes8-4.pdf

Marissa B.

the positive nth root of a positive number x can be written as a power of x, namely, as x^1/n.
the radical sign allows all of the positive nth roots and some other numbers to be represented.
for any nonnegative real number x and any integer n is greater then or equal to two, n square root x = x^1/n.
for all positive integers m is greater then one and n is greater then or equal to two, and all nonnegative real numbers x, n square root x^m= (n square root x)^m= x^m/n.

Section 8-5: Products with Radicals

Notes: aanotes8-5.pdf

Sarah P.
Root of a Product Theorem
For any nonnegative real numbers x and y, and any integer n> or = 2, (xy) ^ 1/n
In the power form, it would take x to the 1/nth power and y to the 1/nth power and multiply them together.
In the radical form, it would take the nth root of x and then the nth root of y and multiply them together.
This theorem is used to rewrite expressions involving radicals or to recognize alternate forms of answers.

To simplify an nth root, we factor the expression under the radical sign into perfect nth powers. THEN, apply the Root of a Product Theorem.
We also learned about means. We already know about an arithmetic mean; When you add the numbers in a set and divide the sum by n. It is also called the average.We learned what a geometric mean is. When you multiply the numbers in the set and then take the nth root of the product.
Geometric means can be used when numbers are dispersed. It keeps one large number from affecting the measure of center.

Nate F.
In this lesson, we learned how to simplify radicals and the things in the radicals. One theorem in this lesson is the Root of a Product Theorem, which states for any nonnegative real numbers x and y, and any integer n> or =2, (xy)^ 1/n. To simplify, (factor) the radicals, you must find everything that can equally be divided depending on the root. If nothing can be broken down, you must look for factors of the number or variable that can be rooted. After that, you have to put the divisible numbers and variables in one quantity, and the non-divisible numbers and variables into another quantity.

Section 8-6: Quotients with Radicals

Notes: aanotes8-6.pdf

Max J S.
In this lesson we learned about quotients with radicals and how to simplify them. We learned what it is to rationalize the denominator of a fraction. This means rewriting the fraction without an irrational number in the denominator. We also learned that this can be done by multiplying the fraction by the irrational number (that's in the fraction's denominator) divided by itself. This can also be done the same way that we got rid of imaginary numbers in the denominator of fractions.

Shannon M.- Rationalize the Denominator- Rewriting a fraction without irrational numbers in the denominator
For example, to rationalize the number 3 over the square root of 2, you would multiply the numbers by the square root of 2. To get rid of a radical, multiply the numerator and denominator by the radical n times, where n is the root that is expressed.

Section 8-7: Powers and Roots of Negative Numbers

Notes: aanotes8-7.pdf

In this lesson we learned how to solve powers and roots of negative numbers. In the equation (-x)^n if n is even the answer will be positive, and if n is odd the answer will be negative. n can be any real numbers.( Positive, negative or zero) Remember that all properties that we have studied before will work when the base is negative! Lets look at an example.
1.) (-3)^4 * (-3)^-6 = (-3)^-2 = 1/(-3)^2 = 1/9 *You can only add the exponents if the bases are the same*
The square root of x is real, the square root of -x is imaginary, x is greater than or equal to 0.
When x and y are negative, the square root of x times the square root of y does not equal the square root of xy.
We can take the nth roots of negative numbers. When x is negative and n is and odd integer > 2 the real nth root of x is the nth root of x.
The nth root of a negative number, x tho the nth root when n is even is not defined.
Theorem: When the nth root of x and the nth root of y are defined and real numbers the nth root of xy is also defined and the nth root of xy = the nth root of x times the nth root of y. We cannot do this with even roots of negative numbers.
-Kaitlynn R.

Section 8-8: Solving Equations with Radicals

Notes: aanotes8-8.pdf

In this lesson we learned how to solve equations with radicals. When working with equations with radicals you always need to be checking your work or you might find an extraneous solution. An extraneous solution is a solution that is found when solving an equation with a radical which is not a solution to the original problem, in short you might get a wrong answer, so check your work and if you did everything right you might find that there really is no real answer to the problem. We also went back over the distance formula in this lesson which is external image dist07b.gif (The lesson said the sub 1s are first, but it does not matter) the distance formula can lead to equations involving square roots. This formula may look complicated, but they can be solved in the same way as simpler equations are solved.
~Austin R.