Question 1

Hi do real numbers include negatives? Because I can see you mentioned in the X can be any real number.

Accepted Answer

Yes, real numbers include negative numbers. Real numbers are complex numbers whose imaginary component is 0. In other words, any number that can be placed anywhere on the standard number line is a real number.

Question 2

What does `such that` mean in mathematics?

Accepted Answer

The term "such that" tends to stand in for "in a way so that" or "in order to". For example, if you were asked to make a liner system "such that" the lines were parallel, it would mean you would make a linear system with the graphs being parallel.

Question 3

what is domain in exact explanaition

Accepted Answer

In its simplest form the domain is all the values that go into a function, and the range is all the values that come out. Sometimes the domain is restricted, depending on the nature of the function.
f(x)=x+5 - - -  here there is no restriction you can put in any value for x and a value will pop out
f(x)=1/x - - - here the domain is restricted since x cannot be zero because 1/0 is undefined. So the domain here is all number except zero.
f(x)=√(x-5) - - - We cannot take the square root of a negative number, so x must be greater than or equal to 5 since for x=5 and up x-5 is positive. In this case, the domain is all numbers greater than or equal to 5.

More here:
http://www.mathsisfun.com/sets/domain-range-codomain.html

Question 4

In the domain and range definitions, how are skips in x written? For example, if x is more than -5 but less than 1, and more than 1 until 5. So 1 is skipped. I see how it is written in the function definition itself, but how is it written in the domain and range, which accounts for the entire span of possible outputs for all inputs? In the past two videos Sal has given examples where the functions are continuous, but not when there are skips. Even in this video where Sal combines the spans of three outputs, the spans overlap, and there is no skipping.

Would this be written as follows: ...-5≤x≤5, x≠1...? -domain example
Or something like this: All real values such that -5≤f(x)<1 or 1<f(x)≤5? -range example

Would this be written as follows: ...-5≤x≤5,  x≠1...? -domain example
Or something like this: All real values such that -5≤f(x)<1 or 1<f(x)≤5? -range example

Accepted Answer

That is a great question.
Typically if the domain is something like all values between -5 and 5 inclusive excluding 1, we write:
χ ε [-5. 5]/{1}
Basically, we write x epsilon non strict lower bound -5 and non strict upper bound 5 and then a slash and 1 in curly brackets. Likewise, if we have the domain is all real numbers except 5 and 7, we can write:
x ε R/{5, 7}

Question 5

it is has hard to grasps concept of each video because there is seems to be that there is no objective is mention/discuss in each video except title but that is not enough and what is context or what we are doing?

Accepted Answer

If you are jumping around from video to video, that may seem like the case, but if you are following the order presented, each video builds from the previous.
Keep Studying!

Question 6

around 4:45 why does Sal put -3 on the left side? Shouldn't he put it on the right

Accepted Answer

First you need to realize that < g(x) < is saying that the leftmost number needs to be less than the rightmost number.

For this part we start with the function 1-x and the two endpoints are 3 and 4.  If we solve 1-x for these two points we get -2 and -3 respectively.  so even though to start 3 si less than 4, when we solve 3 gets us the larger result, at -2.  Similarly 4 gets us the smaller result, at -3.

So to keep it in order we need it to look like -3 < g(x) < -2  Does that make sense?

the reason this happens is because in 1-x, we are subtracting x, so it does the reverse of the numbers being plugged in.  4 is greater than 3, but since you are subtracting you are subtracting more , making the result smaller.  I also hope that makes sense.  if not the main takeaway is to plug in each end point and see which is bigger and smaller.

Question 7

Whats the point of watching this. There are no exercises after it.

Accepted Answer

He's trying to show you a different way of thinking about and simplifying functions.

Question 8

Shouldn't all functions have a unique output? Then why is the function Sal took as an example still valid when both -3 <= x <= 1 AND 1 <= x <= 4 overlap with -3 < x < 4?

Accepted Answer

*NO*, a _function_ does *NOT* have a unique output.

A _function_ have *one and only one* output for each input. There is nothing to do with uniqueness because it could have the same output for different inputs.

Example: `_f(x)_ = |x|`
if we input `_f(5)_ = 5` and `_f(-5)_= 5`.
you see here we have the same output for different inputs and that's okay.

Question 9

I clearly understood 'Piecewise function', the only the only thing I'm stuck at is the application of this concept in real-life situation 
I meant where this concept is used in 'Daily Use'

Accepted Answer

Postage is often a good example.  For a weight of 1-5 pounds, it costs one amount, then 5.1-10, a second amount, etc.  Also, sometime when you are buying things in bulk, groups of amounts that you buy will cost the same and then goes down the more you buy (this is particularly true when a store buys things at wholesale costs).

Course: Algebra 1 > Unit 10

Worked example: domain & range of piecewise linear functions

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Video transcript