Graphing Inverse Functions

Last Lesson, we learned about inverse functions. By now, you should have a basic idea of what an inverse function is and how to solve an inverse function. Today, we're going to go deeper. We're going to try to map out an inverse function on a graph!

To make the most out of the lesson, please make sure you are comfortable with interpreting and plotting simple linear graphs.

Alright.
 Look at this graph...or graphs

Pay attention to the blue line. Ok. Now, lets work out the inverse of the blue function f(x)=2x+1...If you do not know how to do this, go back to the last tutorial and LEARN !!! You're going to be seriously confused if you don't read that post! Alright...Enough ranting, lets solve.

Use the algorithm I told you guys about in the previous post.

1. f(x)=2x+1

2. y=2x+1

3. x=2y+1

4.  2y=x-1
   y=(x-1)/2 (Divide each side by 2 to make 'y ' the subject)

5. f^-1(x)=(x-1)/2

Now, are answer is in fact, the red line! If you plot them onto a graph, you get two linear equations, and yea, I think the graph gives it away, the inverse is basically a reflection in the equation y=x !!!

So, next time you plot a graph....if you want to map the inverse on a graph, simply draw the line y=x (shown above), and reflect all your points in the original function in the line y=x to get the graph of the inverse!.


Magic.

Introduction to Inverse Functions

When you get to pre-calculus, you might have to learn inverse functions....What is that? It's basically, a function that undoes another function.Yea, what does that mean? Well, let me show you how it is done and maybe then you will have a better understanding.

lets take the f(x)=x+1. We want to find the inverse.

YOU SHOULD BE COMFORTABLE WITH REARRANGING EQUATIONS

Follow these steps:

1. set f(x)=y

2.y=x+1

3.swap the x's with the y's.so, in this case, it'll look something like this

      x=y+1

4. Rearrange to make y the subject.

  Lets do it: y=x-1

now...functions are denoted as f(x)...inverse functions are denoted as f^-1(x)

so your final answer will be f^-1(x)=x-1

The highlighted part is the most important, so be sure to memorize it!

Lets do another problem with the Step by Step approach:

Find the inverse of f(x)=3x+2

1. f(x)=y

2. y=3x+2

3. x=3y+2 (I swapped the x's' with the y's, its that simple)

4. Rearrange---

x=3y+2

x-2=3y+2-2

x-2=3y

(x-2)/3=3y/3

y=(x-2)/3

so your answer is f^-1(x) =(x-2)/3

A simple introduction...Later, we're going to learn how to map out inverse functions, graph them, and more.

Multiplying functions

We know how to add functions such as X+1 and X+2.

So,how do we multiply these functions?
Don't, mind me saying, its actually dead easy.

so, lets take a function. (A+1). And another function we want to multiply...lets say (A+2)...A is basically any number...

Skills required to continue:
1.2nd grade multiplication ability...
2.Ability to treat letters as numbers...
3. You know what powers are.

 So, lets first set up our problem.

(A+1) x (A+2)
However, conventionally, we can omit the 'x', so the question simply becomes:
(A+1)(A+2)

now...
WATCH, as I unravel the secrets of multiplying this out.

It goes like this
(AxA+2xA+1xA+1x2)

First multiply the first A by the first A in the second function.
Then multiply the first A by the 2 in the second function.
Now, multiply the 1 in the first function by the A in the second function.

Lastly, multiply 1 in the first function by the 2 in the second function.

So, basically, each term in one function has to multiply each of the numbers in the 2nd function.

Sounds complicated.

Then, according to the signs, add or subtract...In this case, its addition as all the numbers involved are positive.
So, completing what we had above.
Answer:
(A^2+3A+2)

If your still confused, I bet you will be, i'll post a bunch of exercises and solve them on the spot!




















 

Introduction to Absolute Value

In Highschool and College, absolutes values are so commonly seen. That is why it is important to understand what it is about.

First, what does an absolute value look like?
Something like this:







Yea. So, in order to make the most out of what's next, please make sure you have at least an elementary understanding of functions, algebra...Otherwise, you might end up being confused and thats the last thing I want to happen.


So. The 'X' has a value...It could be anything...6,4,435,43,-45,435,43,5. Doesn't really matter.
So, now you might ask, What on earth are those two vertical lines for, surrounding the X?

I'm glad you asked. What the two lines do are they basically make sure the value of X is ALWAYS POSITIVE

If your going to take out anything from this lesson, its what I just highlighted!!!!

Basically, if X=-2, the two lines will make X=2
If X=-5464564, the X will it positive whatever I just typed.
.....
That's probably even easier than addition...

Next time, we'll try to map the absolute functions on a graph.

1. Well, what happens? You tell me. Wait, if you knew, why would you be here? Oops, Sorry.
Anyway, what do Red  Blood Cells do? They carry oxygen around the body right? How? Oxygen can combine with the Hamoeglobin in Red Blood cells to form what they call, Oxy-hamoeglobin.

So an appropriate answer would be, Red blood cells 'take up oxygen' and becomes oxy-hamoeglobin.Right?

b) What do White Blood Cells do? In GCSE level, they essentially produce two types of antibodies, Lymphocytes and Phagocytes, to protect the body from pathogens, bacteria, harmful stuff. So that's probably what its existence in the alveolus is for. PROTECT THE LUNG FROM PATHOGENS!!!

Radiation---An Introduction

In life, radiation is integrated into three different types, Alpha, beta and Gamma Radiation. You must have all heard of the term 'Radiation', but this post is more concerned about Radiation into specifics. But not too complicated. Just a basic idea of the types of radiation and the hazards each radiation poses. After all, not all of you want to be Radiologists. BUT we do want to be able to save our selves from the adverse affects of technology, hence emphasizing the importance of understanding all this.

 Before we continue, I want to loosely define the term radiation: Wikipedia defines radiation as: radiation is a process in which energetic particles or energy or waves travel through a medium or space

Now we got that out of the way, lets move on into what we really want to know

Alpha Radiation

 The picture on the right is an Alpha radiation model. The blue and red bits symbolize Neutrons and Protons. I will do a separate post about that later on.

Alpha Radiation is a type of 'energy' decay which releases an Alpha particle. This transforms into into an atom with a mass number 4 less and atomic number 2 less, also equivalent to that of a Helium Atom.

The example below is a good illustration of what I'm talking about.

238 92U

→

234 90Th

+

4 2He2+

The Uranium loses 4 mass numbers from 238 to 234 and it also loses two protons; from 92 to 90. If you don't understand what I'm talking about, feel free to check out my tutorials on Protons, Neutrons and Electrons.

Alpha Particles have very little energy. Alternatively, we can say that it has a low penetrating power.. If you zap a ray of Alpha Particles into a sheet of paper, most of the particles will be absorbed. That's how weak it is.

Beta Particles

Beta is the particle in the middle. Beta radiation does not release a Helium Atom, but a Electron, hence making it negatively charged. Take a look at this example.





A neutron is converted into a Proton, and in the process, an electron and an electron anti neutrino is emitted. Lets forget about the electron anti-neutrino, that is hardly elementary, and just remember an electron is emitted, and that beta is NEGATIVELY charged. ok, good.

The 'Penetrating' Power of Beta Radiation is much stronger than that of Alpha, as it can easily penetrate paper. However, it cannot penetrate a piece of Aluminium, which is of an order of magnitude thicker than that of paper!


Gamma Radiation
I am pretty sure most of you have heard of this type of radiation.

Thankfully, Gamma Radiation carries no charge. However, it carries the most penetrating power, being able to comfortably penetrate most of a thick piece of lead. It is a form of electromagnetic radiation which can cause genetic mutations

There is a whole lot more about Gamma radiation, for this lesson, i'll adhere to brevity and simplicity.

Factors of Production

In a business, it is important to consider the factors of production, or commodities used to produce goods.
The 'four factors of production' in a business are:

1. Land
2. Labour
3. Capital
4. Entrepeneur.

1. Land
: is the natural resources and therefore includes not only the land that we stand on, grow crops on and build upon but also fisheries, mines and forests. The Payment given to the owners of land is rent.

2. Labour
: is defined as all the physical and mental effort devoted into the production process. The payment for labour is called salary/wages

3. Capital
: is the manufactured resource that is used in the production process. These are things like machinery, things that can help the business produce things. Payment to capital is interest.

4. Enterprise
: Is the unique ability someone in the business has to organise the factors of production and implement changes in a business to allow the business to be profitable.