How do we recognize functions?

A Relation is a set of ordered pairs. The set of ordered pairs in a relation can be shown using braces, tables, graphs, and mappings. 

A Function is a relation that assigns exactly one output value (range/y-value) for each input value (domain/x-value).

Recognize functions by making sure the same input value (x) is NEVER repeated. 

Some characteristics of a function are:
1. Each x-value must be matched with a y-value
2. Some y-values may not be matched with a x-value
3. Two or more x-values may be matched with the same y-value
4. A x-value CANNOT be matched with 2 different y-values.

How do we add and subtract rational expressions?

So what is a rational expression?
A rational expression is an expression that is the ratio of two polynomials.
So what does that even mean? Well basically it's a fraction with polynomials.
Hers's an example: 
                              
Okay so now that you know how a rational expression looks like, what happens when you add two rational expressions? 
Well, let's say you have to add (x-5)/(x^2-9) + (x+12)/(x+3), how can you add them? If you had to add 2/5 + 1/5 you would get 3/5 but that's because their denominators is the same, but for our little rational expression above, the denominators are different so what do we do?!?!
Well we actually already mentioned the answer, we must find the Least Common Denominator! 
(But before we go any further, let's look at a picture so we can see the problem with our eyeballs and possibly better understand it!)
                        
So the first step to solve this problem would be to factor out the denominator: (x^2-9) which would become (x-3)(x+3) and if we look at the other denominator it already has a denominator of (x+3) which means that all you have to do to make it equal with the other denominator is multiply it by (x-3). Don't forget that if you multiply that rational expression by (x-3) you must multiply BOTH the numerator and denominator!
Once you do that you should end with (x+5) + (x^2+9x-36) all over (x+3)(x-3) and your final answer should be (x^2+10x-41)/(x^2-9).

How to factor by grouping

When you are asked to factor by grouping you will be given a problem similar to the 

following:

8x^3 - 64x^2 + x - 8

So how do you go on about solving it?

Let's break it down step by step:


1) You split the equation in half and you take the first two terms (8x^3 - 64x^3) "group 

them" and find the greatest common factor between them which would be: 8x^2 (x-8)



2) Do the same for the other half of the equation (X-8) and the greatest common factor 

would be 1(x-8)

3) Now you have something that looks like this:

8x^2(x-8) + 1(x-8)



4) Finally all you have to do is combine the outside of both mini-equations to create another 

equation and times it by the inside. So your final answer should look something like this:

(8x^2 + 1) (x - 8)

Why do we flip inequality symbols?

You probably remember in elementary school when you used to use inequality symbols and the teacher would tell you to think of them as little alligators trying to eat the bigger number.             Ex. 1<10 One is less than ten.

                             
Well we're in high school now, and the inequality problems are no longer as simple :(
This is how an inequality problem looks now: -10x-20<5x+25

So how do you solve this? Let's take a look at it:
Step 1) Subtract 5x from both sides to get -> -15x-20<25
Step 2) Add 20 to both sides to get -> -15x<45
Step 3) Divide both sides by -15 to get -> x<-45

So based on our work, x should be less than -45, so let's pick a number less than -45 and plug it into the inequality to make sure its correct. Let's say we pick the number -46, plug it in -> -10(-46)-20<5(-46)+25 which equals to 440<-205. Woah! Wait a minute, that doesn't seem right! We must have made a mistake in our work, let's go back and check. Aha! It seems that we didn't switch the inequality symbol once we divided by -15.

So why must we flip the sign whenever we multiply or divide by a negative number?
Well let's look at a number line to help us with it:

Okay so lets say you have -2x<10, whenever you multiply or divide by a negative number (in this case we are dividing by -2), it's as if you flipped the number line, so this means that 10 turns to -5 (and -2 turns to 1) and a negative number is obviously less than a positive number. Thus, the inequality sign will no longer be facing the now -5, it will turn around and face the 1. The way i like to think of it is whenever you multiply or divide by a negative number, the little inequality symbol gets scared and turns away! :D

Hopefully you understood this concept, and as a reward for your perseverance and patience in reading this post, here's a picture of a kitty! -->
                            
Follow the kitty's example and take a well deserved nap, they're awesome! :D