Katie's Lesson on Composition of Functions

Composition of Functions

     A composition of functions is when you combine two separate functions so that you use the output from one function for the input of the second function. (Or the range (y-values) of one function becomes the domain (x-values) of the second function. 

A composition of functions looks like this:

                                            

When reading this out loud, you say "f of g of x"

It is important to remember to go in order across from the inside out when using a composition of functions. 

When solving for f(g(x)) you first plug in your g(x) function into your f(x) function. For example, if you were given f(x)=5x and g(x)=10x+3, then your first step would be to plug g(x) into f(x).

This would look like f(g(x))=f(10x+3). Can you see how the function g(x) has taken the place of the x in the function f(x)?

The next step would be to plug the f(x) function into what you already have. So now you take your 5x and put it into the equation.

This would look like f(g(x))=5(10x+3)

                                            =50x+15

Unless you are given an x value to work with, this is as far as you can solve the problem.

You can also reverse the order of the equations and be asked to solve g(f(x))

Given the same functions as above and following the same procedure, you would find that after the first step g(f(x)) would look like g(5x)

In the second step, it would then be turned into g(f(x))=10(5x)+3

                                                                                     =50x+3

It is clear that the answers to the functions will be different depending on the order of the function.

Because of this it is extremely important to make sure you keep the functions in the correct order and plug in using the correct order. 

Let's try another example

If f(x)=4x+7 and g(x)= 3x find f(g(x)) when x=3

     f(g(x)=f(3x)
          =4(3x)+7
          =12x+7
          =12(3)+7
           =36+7

       =43          

                     
Now find g(f(x)

g(f(x))=g(4x+7)
          =3(4x+7)
         =12x+21
         =12(3)+21
         =36+21
     =57      

Now you know how to compose two functions! Good luck!

Beginning Input/Output

This link goes to my power point presentation

https://docs.google.com/presentation/d/132UT8c53JV15zX8-fDnrPjURrD-qV-rom6VB8CqaWoU/edit?usp=sharing

Nicoles be the professor

COMPLETING THE SQUARE:

 

This trick is used to factor quadratic expressions, especially when other methods (like the x-method) doesn’t work:

 

First you write your formula (ax² + bx + c) in the form ax² +bx= -c (move your c to the other side of the equation)

 

Divide b by 2, then square it (b/2)².

 

Add the (b/2)² to both sides of the equation: x² + bx + (b/2)² = -c + (b/2)².

 

Write the left side of the equation as a perfect square: [x + (b/2)]² = -c + (b/2)²

 

Let’s pretend that a=4, b=16, and c=-18

 

Move c to the right side of the equation to get 4x²+16x= 18

 

(-18 becomes positive when you move it to the right side)

 

Divide both sides of the equation by 4: x²+ 4x= 18/4.

 

Take ½ (4) which is the x term, and square it to get (4/2)²=4.

 

Add the 4 to both sides of the equation x²+ 4x +4= 18/4 + 4

 

Write the left side of the equation as (x+2)² which is a perfect square and you get that (x+2)²= 34/4

 

Then you’re done!!

 

Simple as that!

 

Well, not really, but if you do it a bunch, it eventually it will make sense. But only if you have a teacher who is patient and helpful like I do!!

Teaching Average Costs

Hello everyone, my name is Lucy Shepherd and I am here today to teach you about Average Costs. Hopefully with my help you will come to understand why average costs are important in everyday life.

 The average cost is the cost to produce an individual unit of an item based on a certain quantity of objects. Say for instance a company wants to produce 500 (q) computers. Say to build 500 computers it takes $800,000 (C(q)). The average cost to produce 1 of those 500 would be $1,600 (a(q)).

 I know this may seem like a tedious and unnecessary step, but in the long run it is really quite important. Why, you ask?

The graphs of cost functions are almost never straight. The graph of the average cost is going to be a straight graph of a linear function.

Companies that manufacture items need to keep a close eye on what they are spending per item. However, companies are not the only ones that pay attention to average cost. When you go to the supermarket and look at the price sticker on the shelf, the price per unit in the top corner is your average cost for that item. You can also find the average cost for an item in a bulk order. Say you are ordering a bulk of rubber ducks. You decide you like these rubber ducks from Oriental Trading. They cost $13.99 per 26 ducks. The average cost per duck would be $0.54 per duck. If you needed more than 26 ducks and you had a budget per duck (for some odd reason) you could then tell if the ducks were in your budget.

Thank you all for being here today. I hope this lesson gave you an insight into average costs and why they are important.

Dawson Slope Lesson

Here is a link to my lesson plan explaining slope

http://www.scribd.com/doc/220370894/Slope-Lesson-Dawson

Thanks for taking a look!

Be the Professor: blog 4

Hey guys I made this video as a lesson on product rule for derivatives.

http://www.screenchomp.com/t/41i62zb6DTS

This is my blog vid, thanks!

Be the Professor: Blog #4

Hello class! I’m Professor Jenna and today’s lesson is about domain and range. Domain and range are very important when interpreting a graph or understanding the x and y values of a function. You will need this information to analyze graphs and whatnot. 

So, what is domain? Domain is the set of input values that go into a graph or function. These points define the function. And what is range? Range is the set of all the output values of function.

For example, the surface area of a cube is the total outside area of the cube. It can be found by using the formula: A = 6x². Let us assume that you are asked to construct a cube that has a side length of at least 3 inches. In order to identify the domain, you must first identify the independent variable. Since the independent variable is the length of a side, then the domain has to describe numbers that could represent lengths of sides for this problem – the problem says the length has to be at least 3 inches, so the domain is all numbers greater than or equal to 3.

In order to identify the range, you must first identify the domain. The range is the y-value that corresponds with the chosen x-value in a function. So, to find the range, you simply find the area of a cube with a side of 3 (the smallest one you can have), and that tells me that my range has to be greater than or equal to 54.  

There are different notations for identifying domain and range. There is the set-builder notation and the interval notation:

Set-builder:

 

Interval: ( a, b)

Also, for those of you who are more visual learners, here is a graph with the domain and range specified after: 

Domain: (negative infinity, infinity) because the graph stretches out on the x-axis.

Range: (-3, infinity)

If you have any questions please let me know and have a wonderful weekend!