How do we perform compositions of functions?

Compositions of Functions:
     Example: If f(x) = x^2 and g(x) = x+6
Find f(g(4)).
   g(4) = 4+6 = 10
f(10) = 10^2 = 100 .

Compositions with Variables:
   Example: If f(x) = x^2 and g(x) = x+4
Find f(x+4)
    f(x+4) = (x+4)^2
(x+4)(x+4)            (Distribute)
      x^2 + 4x + 4x + 16
x^2 + 8x + 16 .

The term "composition of functions" (or "composite function") refers
to the combining of functions in a manner where the output from one function
becomes the input for the next function.

In math terms, the range (the y-value answers) of one function becomes
the domain (the x-values) of the next function.

The notation used for composition is: and is read "f composed with g of x" or "f of g of x".

(Regentsprep.org)

How do we identify one-to-one fuctions?

A function is said to be one-to-one, if for each element of the range, there is a unique element of the domain. (for every y there is one and only one value x). if whenever
f (a) = f (b) then a = b.

Lets try out these problems,

1. {(3,2), (1,7) , (0,2)} this is a function but it is not a one-to-one function.
2. {(2,1) , (4,2) , (1,4)} this is a function and it is also a one-to-one function.

(regentsprep.org)

"One-to-One"

NOT "One-to-One"

How do we use inverse variation?

First, lets list ALL the interger dimensions for rectangles whose area is 24
                            L           |         W
                               1                 24
                               2                 12
                               3                  8
                               4                  6
                               6                  4
                               8                  3
                              12                 2
                              24                 1
This relationship is called an Inverse Variation.
We say that x varies inversely with y.
       xy = 24                       when x goes up, y goes down.
       y = 24/x                      when y goes up, x goes down.

If x varies inversely as y and x = 12 when y = 3, what is the value of x when y = 9?
                     xy = k                            
12*3 = k
k would equal 26

x*9 = 36
  9        9
x would equal 4

How do we solve exponential fuctions?

Well we already know how to solve exponential equations, or at least we get the hang of it.
     Our new challege is to solve exponential functions.

The general form of the exponential fuction can be written:
     f(x) = ab^x
this is where a is the initial value, and b is the growth rate.
     w(t) = 100 * 2^t

Lets try this problem;
     A population of rabbits doubles every 60 days according to the formula P = 10(2)^t/60 , where P is the population of rabbits on day t, what is the value of t when the population is 320?

a ; would be the initial population, where b would be the growth rate.
          320  = 10(2)^t/60
           10            10

          32 = 2^t/60
         2^5 = 2^t/60
       60*5 - t/60*60    (cancels out)
t would equal 300

How do we solve exponential equations?

Well usually we know how to solve problems like:
          x = 3^4
which will equal 81.

We know how to also solve problems when the unknown is alone:
          (3)^3 = (x^1/3)^3
which will equal; x = 3^3 = 27

We know how to solve problems where even the base is unknown:
          4 = x^2
which will equal to -2,2.

What we didn't know is this;
          100 = 10^x , but this problem is easy, x would have to equal 2 in order for this to be true.
this problem is where the unknown is actually the exponent.

          Solve for x;
2^11 = 2^x
which will equal 11.
          When the bases are the same, the exponents have to be equal as well.

Solve for x:
          2^x-3 = 4
        2^x-3 = 2^2
            x-3 = 2
            +3=+3
    _____________
Final Answer, x=5

Aim: How do we multiply and divide rational expressions ?

Multiplying and dividing may seem difficult, but when using them to solve rational expressions, it can be a lot easier. First, we start out by factoring out all of the numerators and the denominators so we can make things much easier for us. Then we get to cancel out all of the common factors to simplify the expressions. Lastly we have to multiply the remaining factors in the numerator & the denominator. We can find out the answer with these simple steps.
   Problem:

2x +10x + 12x          x + 7 
____________  *     _____      
    
    2x + 10x               x² − 9

                        ↓

2x(x +5x +6)       x + 7 
__________  *    ________            (Factor it Out)
2x(x + 5)          (x+3)(x−3)

                   ↓

 2x(x+2)(x+3)           x + 7 
____________   *    _______             (Cross Out)
2x(x + 5)              (x+3)(x−3)
                   ↓

 (x+2)          (x +7)         x² +9x + 14    (Multiply Remaining Factors)
______   *   ______  =   ___________

 (x+5)          (x−3)          x² +2x − 15

  
                   ↓

x² +9x + 14
__________                (x cannot equal; 0, 3,−3,−5)
x²  +2x − 15


DIVIDING:
   First you start out by using KCF (Keep, Change, Flip) which is the multiplication of reciprocal.
  Problem : 
x + 7                   x² + 9x + 14                        x + 7            3x² - 9x
____        /      ___________        KCF        _____    *   __________
x² − 9                  3x² − 9x                             x² − 9       x² + 9x + 14

Step 2: Factor out everything that needs to be factored.
     x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 3: Cancel out the common factors

    x + 7                 3x(x − 3)
_________     *     ________
(x− 3)(x+3)          (x+7)(x+2)

Step 4: Multiply the remaining factors of the numerator & denominator.

      3x                        3x
_________    *    ____________
(x+2)(x+3)           x² + 5X + 6

Final Answer

    3x
__________     (x cannot equal;  0,−2,3,−7,−3)
x² + 5X + 6


http://www.mathwarehouse.com/algebra/rational-expression/how-to-multiply-divide-rational-expressions.php

Aim: How do we solve problems with the rational expression operations?

For starters, Rational Expressions can be defined as fractions that have variables in their denominator.

Multiplying Rational Expressions: We should begin by searching for things that can be eliminated otherwise known as factoring it out. Once you have that done, you can start to divide out the common factors and simplify the expression. For Example ;



 -> You look to see if you can factor anything out.
Nothing can be factored out just yet.
 -> Divide the common factors.

= 

-> Simplify.


Final Answer will be