| Start | So, a lot of what we do in the first
two weeks or so of the course
is really review.
It's really just designed to make sure that you know your basics of algebra functions, etc. before we start getting into real calculus. Okay? Um, so last time we just talked about uh, like what are graphs? Those kinds of things. Now we're gonna do a little more on functions. |
| 0:31 | So, math is all about functions.
That's not really true, but when we deal with math everything we deal with is going to be some kind of function. So mathematicians are very picky about this stuff. They wanna know that the, what you're doing when you're applying some operation to something that there's a set of rules that will always work because if there's exceptions, then you haven't really got a perfect rule and that means you can't say that things are always true. You know that. Your parents would say something, you'd look at them you'd say that's not true. |
| 1:01 | You give them an example and they just get annoyed.
Because you're supposed to be the kid and they're supposed to be the adult. So, in math there's, you wanna create things where you can say under these conditions this will always work and this will always be true. So, a lot of what we deal with are functions So functions have specific meaning in math. But a function in general is anything that takes an input and creates an output. Sometimes functions do nothing. Okay? |
| 1:30 | Uh, so eating we take in food you excrete whatever you excrete
and you get by energy so the input is food and the output is energy.
Okay? Um, you can think of how you have functions that involve your Stony Brook ID. Okay, you type in your Stony Brook ID and then you get out whatever it is that you were doing with your ID. Um, so in math the input we usually call x, but we don't have to. And the output we usually call y. But a function |
| 2:01 | Is you have an input
you apply the function
and you get the output.
And as I said, the function could be anything. Okay, it doesn't have to be a magic box. Usually when we're dealing with them it's gonna be an equation. So you take some number you put it in and you get output. What we call the input is the domain. And what we call the output is the range. |
| 2:33 | That's as good as my handwriting gets folks.
Don't get excited. Okay? So if you have a function like f(x)=x^3 Okay? You'd put in the number and you'd get out the cube of the number. Right? So the function would be x^3. So for example if you put in |
| 3:01 | 2 the function is to cube it and you get out 8.
Okay, this is very, very common sensicle. But with functions it's very important that we establish a certain rule about what we can call a function and what we don't. So one of the things about functions is you wanna make sure that whatever you put in there's only one possible thing you can get out. So we have a way of defining that in a function we say |
| 3:41 | And I'm gonna play with this in a second, but for every x
remember x is in the input
there's only one y.
Now, we could put in different x's and get the same y. But when you put in an x you cannot get out different y's. Okay? So again, whatever you put in x you only get one output. |
| 4:05 | And to make this more broadly in math speak this class isn't gonna get too hung up on that, but it's important.
? For every element of the domain, also because I'm doing it on video Okay? So that's x |
| 4:45 | Okay, so for every element of the domain, there is only one element of the range.
Not necessarily the other way around. If you wanna think about this graphically because usually that's easier When we graph we typically put the domain along the horizontal axis |
| 5:04 | so these are our x values and outputs are here, which we either call y or f(x) so it says when we put an x, even if you put in the same x you only get one y so you get these y values so if you put in a, b, c, d, e, f those are your x's that go in |
| 5:32 | okay, you get a unique y so you don't get
different y values you only get one y value.
Okay? So notice, say a, b, and e all have the same y value that's okay. Because you can put in different x's and get the same y But what you don't get say it's something like that where when you put in c you get two possible outputs. Because there's a way to test that and that's called the vertical line test. |
| 6:01 | So if you look at a graph It says, um, there is no |
| 6:44 | So way mathematicians is fun is they say what do you mean by intersect what do you mean by graph
what do you mean by place- they're really fun.
But, to be not super technical there's no vertical line that intersects the graph in more than one place. It can intersect it in zero places or one place, but not more than one. |
| 7:05 | For example
a graph like that
Okay, no matter where you draw a vertical line
you're only gonna hit the graph once.
You can't go through the graph a second time. |
| 7:31 | You can do a horizontal line, but that's not what we're testing. We're testing a vertical line.
And if you had a graph that looked like this and if you draw a vertical line right there you can intersect it at three places, there's probably a couple of spots where you could get it in two places so this is not a function. Not a function of x, which is all we're concerned about. And this is a function. A little later when we talk about inverses, inverses also have to pass the horizontal line test. |
| 8:05 | But for now, so if you look a graph all you have to do in your mind is imagine that you're drawing a bunch of vertical lines
and say to yourself do I hit the graph 2 times? or more than once?
If the answer is yes then it's not a function, and if the answer is no then it is a function. Even if it only does it in one spot it's not a function. Because remember the function says there is no vertical line. |
| 8:33 | So you can have something called a piece wise function.
A piece wise function is a function that is designed in pieces. So say you have this and you have that. Well this is a function right up to a. And this is a function right up to a. Unfortunately they're both valued at a. |
| 9:04 | Your line can cut through right there
So since a line would go through the graph at a, so in other words when you plug in a
you don't know what answer to get because you say well I get two different answers
so I'm confused.
Okay, you know, is it true or is it false? Is it a? Is it b? Is it yes no? So the problem is this would not be a function, but if you made this |
| 9:39 | Now if the graph looked like this at a it has the value from the lefthand graph but not the righthand graph.
These are called piece wise functions which you'll see a little bit later in the class. The other big thing about functions that we're gonna care about is domain. |
| 10:03 | That sneeze has been recorded forever on video.
Anybody else wants to get recorded nows your chance. Um okay so domain So domain is essentially what you're allowed to put in for x. Or more generally, allowed values. that you can put into the equation. |
| 10:30 | We're generally gonna deal with x goes in and y comes out, but not always, you know
t can come in and w can come out it really doesn't matter what the letters are.
Okay, but values that are allowed as inputs okay? And then the range would be the values that you get for output based on what you did for input. |
| 11:08 | So in this course we'll spend a lot of time on domain, but almost no time on range.
|
| 11:31 | So remember the range depends on what you put in.
So if you're not allowed to have 10 in the domain then you can't ever get an output based on putting 10 in the equation. That make sense? Alright so let's go through a couple of domain things. |
| 12:02 | Ah so say we have the equation
f(x) = √x-8
or y is √x-8.
Okay, so what's the domain of that? It's not equal to 8. |
| 12:31 | So 9 would work, right?
You could put a 9 you'd get √1. But if it's not = 8 we could put in 7. and get √-1. Which is bad if we're dealing with real numbers, that's not quite the domain. But you're on the right track. Yes. x > 8. x > 8 that's pretty good, right? So because you can't put a number < 8 because you'd get a negative value. What's the √0? 1? 0. You meant 0. |
| 13:00 | Okay, so you can take the √0 right?
Because 0*0=0 So the √0 is 0. So x ≥ 8 would be better. So the domain is x ≥ 8. So since what you do is you take what's inside the radical called the radican, which you don't care and that's x - 8 ≥ 0 and then you solve and you get this. |
| 13:38 | So if I gave you f(x) = √7-x To get the domain you would take 7-x |
| 14:00 | and you would say ≥ 0, which means x ≤ 7
And you could always test it out get your domain and then try values, so you have this domain
x ≤ 7 you say well 0 < 7
you plug in 0 and get √7 that's okay.
It's not a simple number it's an irrational number but that works. And you say well what about 10. Well 10 wouldn't work because you get √-3 so you know you have the right domain so you get the inequality backwards this is a way to test it. |
| 14:30 | Now sometimes you have to be able to do these in what are called interval notation.
And I haven''t played enough with mymathlab, but it's possible that when they give you a question like this that you have to plug into the computer in interval notation. So let's learn that before we do anymore domain stuff. So interval notation is a way of representing the interval in a more compact form. |
| 15:06 | So if you wanna do the number line and you wanna go from a to b
you represent that
with (a,b) in parentheses, that says start at a and finish at b but
don't include a or b.
|
| 15:30 | So this is
not including the endpoints.
That's pretty simple. What if I do wanna include the endpoints? Well then I do a square bracket. Now this includes the endpoints. |
| 16:10 | So say I do some problem and I get that the answer
is, you know, x is less than or equal to 4
greater than or equal to 4 and less than or equal to 7.
? remember you did these you have the solid dot |
| 16:30 | line in the middle and that says
and that says any number between 4 and 7 including 4 and 7.
You would do this in interval notation that way. So what'll happen is you'll do a problem in MyMathLab you'll get the answer that x is between 4 and 7 and you'll have to put it in this way. Okay, I'll play around with it this weekend and make sure. |
| 17:03 | Supposed instead
we can include 4, but not 7.
So we have like that. Then you would use a square bracket on the left and a parenthesis on the right. |
| 17:31 | Straight forward?
Not too tricky? So how would we do x ≥ 8 in interval notation? Well, we can include 8. But what's our endpoint? Infinity. Infinity always gets parentheses. Never gets a square bracket, always gets the soft one, no exceptions. |
| 18:00 | Okay what if I wanna do x ≤ 7, then I would go the other way.
And I would do negative infinity. So far so good? How do we feel about these? Not too scary? Alright let's do some more domain stuff. |
| 18:35 | We can make these harder.
I know you don't want me to, but I can. Say I had What's the domain of that? 5 over x-3. |
| 19:05 | x can be anything except 3.
So if it's the exam you could write all reals except x = 3 or you could write x ≠ 3. How would you do that in interval notation? It's a little tricky. |
| 19:30 | Let's clear some space.
So let's think about the number line for a second. So basically we can do from negative infinity all the way up to 3. Then we kind of have to stop. So if we're using parentheses, right, we would have negative infinity up to 3 but then we can't include 3. But then we also wanna go from 3 on to positive infinity |
| 20:01 | and then the or symbol in math
Is a U like that, okay?
And that's a symbol that'll be in MyMathLab It's not the letter U it's a large upside down U with no little tail or anything on it. Okay, this stands for 'or' That stands for 'and'. I have no idea why. |
| 20:31 | Well, the right side up U s tands for 'union'.
And the upside down is intersection so maybe that was an 'and' Maybe, I don't know. Sure. Okay Sometimes these things were in German, but 'union' in German is 'union'. Union I don't know how to say it just actually, but it's probably something clever like ? with a German accent. Anyway, so that's how you do ≠. |
| 21:01 | Now we'll make it fun.
What if I said f(x) is 5 over √x-3. It's a little trickier but not a lot. |
| 21:30 | Any ideas?
Go ahead, hit me. What do you think? Well, if it wasn't the 1 over it, if it wasn't the square root what would it be? What would the domain be if you just had this? Right. Okay it would be x ≥ 3. |
| 22:00 | But you're in the denominator so you have a problem in the denominator. You can't have 0 down there.
Is that what you were gonna say? Well if you were, that's if you want to play with it you would rationalize by multiplying the top and bottom by √x-3 So what's under the radical has to always be ≥ 0 but you can't have 0 because you can't have 0 in the denominator of a function. |
| 22:31 | of a fraction.
Okay, you could have a 0 in the numerator of a fraction but you can't have 0 in the denominator of a fraction You've never seen a fraction that looks like this. Okay, you might see a fraction that looks like this. This says we've cut the pizza into 5 slices and you get 0. Okay, which is my life, okay but But this doesn't really make any sense because this says we cut into 0 pieces and you get 5 of them. Which really doesn't work here. Remember when you did fractions in forth grade? |
| 23:02 | Okay so the domain
would be x-3 > 0 where x > 3
Or in interval notation 3 to infinity.
So far so good? Alright now we'll really mess you up. |
| 23:55 | Alright take a second and see if you can figure that one out.
|
| 25:40 | We're done thinking? We're ready for me to do it? Go ahead.
Correct, so for those of you who didn't hear him, If two things- first you look at the radical and you say x-2 has to be ≥ 0 so x has to be ≥ 2. |
| 26:04 | Then you look at the denominator and you say but I can't have 5 in the denominator
So x-5 ≠ 0.
So x ≠ 5. So you have x ≥ 2 and x ≠ 5. So if you wanted to do that in interval notation You would say well let's see I can start at 2 |
| 26:30 | I get up to 5 I don't include that
and from 5 I got to infinity.
Now last time I did this somebody said but wait that's an 'or' symbol there but you just wrote the word 'and'. So these are doing two different things Okay? Just so you don't get confused. You can write the word 'or' if you want. In the computer you might have to use the 'or' symbol, but for me you can write that. So a number that satisfies this has to satisfy two conditions. |
| 27:02 | It has to be ≥ 2 and it has to be ≠ 5 so say 3. Yes?
Pretty much never we'll use the 'and' it almost never will show up. Here it will show up. I'm not gonna punish you if you wrote 'and' or 'or'. And I'll tell the TA's not to get excited about that. If they do I'll say it's okay. just getting to here makes me very happy. Math, remember, math is very specific. 'And' has a meaning, 'or' has a meaning. |
| 27:30 | So you say to yourself, well what would work? Well 4 would work because 4 is bigger than 2
and 4 ≠ 5 so 4 goes in this region.
100 would work because 100 is bigger than 2 and 100 ≠ 5. So 100 goes in this region. So you could be in this region or you could be in this region. Satisfying this and this, okay? But generally I don't really care about that stuff. That's where it starts getting a little to mathy. But the important thing is this |
| 28:00 | is what you're looking for.
So now why is domain important? Well when you're doing calculus you just wanna make sure that if you're doing something that it's allowed. Other than that domain isn't really gonna effect you guys that much. You see it in the business context all the time So um a lot of what you're gonna do when you take Business 220 which is, um, decision making and operations research and beyond one of the things that you do is something call optimization. So you're a factory and you're manufacturing cookies and you |
| 28:35 | and you want to try to find the optimal mix of the double fudge oreos vs. the triple stuff
vs. those yummy white chocolate covered ones that you only get at Christmas time.
You have to stock up on it if you want them during the rest of the year. Not that I know about that. Okay um, and you have to decide the optimal mix of how many you make vs. how many you can sell. Okay and your inputs are your flour, sugar, butter? I don't think there's butter in oreo cookies, but whatever they use that pretends to be butter. |
| 29:05 | Okay? And then the output are yummy cookies.
So the domain would essentially be what you're allowed to put in the cookie. Or more realistically in the equations that you would set up in your Excel spreadsheet You put in half a dozen input factors you run a little program and out comes how many of each type of cookie you should make. It's really pretty cool it's a thing built into Excel that does that. |
| 29:33 | Okay and it'll show up in other aspects of business
yeah I guess it'll show up in marketing.
I'd have to think about it for a bit. Alright so let's do another one of these. Go ahead. Yes, hi. I did ≠. |
| 30:01 | Did I do ≠?
Yeah ≠. So x-5 ≠ 0. Oh because you can have < 5 so if you have a fraction right say you have 3 over x-5 you can put in a number less than 5 you'll get a negative answer you can put a number bigger than 5 you'll get a positive answer you just can't put in 5. Okay so the only problem here is x cannot be 5. |
| 30:30 | But x can be 4.9999 it can be 5.01 it just can't be 5.
Good, alright. Let's try one more slightly nasty example. Just to make you feel miserable before labor day weekend. It is a 4-day weekend it's very important that you guys enjoy yourselves. Isn't it great we get that extra day we get that Tuesday? You know why you have to balance the Monday Wednesday and Tuesday Thursday classes. |
| 31:01 | so if we had class on Tuesday, we'd have to get a Tuesday off somewhere else.
Which is fine by me because Halloween is Tuesday, for example. Or Tuesday before Thanksgiving, but we get this one instead. I think we can take all of those Tuesdays. But I'm not in charge, but I don't teach on Tuesdays so it's easy for me. Alright, so what if I gave you |
| 31:33 | that one. Find the domain.
|
| 33:32 | Notice this is upside down from the last one right? I put the radical on the bottom.
Okay so let's look at the bottom first. So we did one like this before, but we didn't have the thing on top, but we still know that 3x-2 has to be > 0 It can't equal 0 because it's in the denominator of the function So it can only be > 0. So this is 3x > 2 |
| 34:03 | x > 2/3
So we have part of this done. What about the numerator?
Yes Hm, well can you have < 11? Because you have a negative number on top? But you can have > 11 right? Because then you get a negative number. Well what's wrong with having 11? Nothing, okay so that's why this is a nasty question. |
| 34:32 | So you can put any number you want up top
because you can have 0 in the numerator so the other type of problem
right, you can't have a 0 in the denominator
but you can have 0 in the numerator so you don't have to do anything.
So that's the answer x just has to be > 2/3 Or in interval notation you get that. So let's just make sure we understand one of these because I don't know they tend to show up on exams. Questions that look like that. |
| 35:02 | Since I write the exams I'd say
you should pay attention to those kinds of- wait was my microphone on just now?
Oh boy You guys heard that. So we could have something that looks like this. Or we could have something that looks like this. |
| 35:47 | Okay so if we look at the left one
the numerator
has to be ≥ 0.
|
| 36:00 | Yes
Oh yes
You're fast, right, so x has to be ≥ 1/2
but x ≠ 4.
See if you're not sure between the 'or' and the 'and' you fudge it and use something like a semi colon or a comma. |
| 36:30 | I always did that and look where I am today.
So here we have two restrictions. One is the thing in the radical has to be ≥ 1/2 and the numerator can't be 0. Now we go the other way around. Here the thing in the radical has to be > 0 But we don't care about the numerator so for this one the answer is simply x > 1/2 |
| 37:05 | Those were fun, weren't they?
How are we feeling about this whole domain thing? You feel it's under control? Do you want one more practice problem? One more? Alright. |
| 37:34 | Then there's other things you can have with domain and range which will show up. Exponentials and logarithms
we don't do sin, cosine, and tangent stuff like that
By the way what if I gave you cube root instead of square root?
Can you take the cube root of a negative number? Remember the rules? Yes, so you can take the odd root You can take the odd root of anything |
| 38:00 | but you can't take the even root
okay so square root, forth root, sixth root
what's inside has to be ≥ 0
cube root, fifth root, seventh root
can be anything you want.
So find the domain so a test question could look something like this. |
| 38:33 | There you go.
It's moderately hard but not too hard. |
| 40:30 | Alright so two things on this problem.
So first, what about the numerator? Almost. I heard less than. Okay, so the numerator has to be ≥ 0. or the thing inside the numerator has to be ≥ 0. So 5 - 3x has to be ≥ 0. |
| 41:09 | So x has to be ≤ 5/3.
So watch your inequality signs. And to test it ask yourself, does 0 work? 0 is always a good thing to test because it's easy to do. So 3 times 0 is 0 so you get the √5 the √5 is allowed. So 0 would work so you can be < 5/3. |
| 41:30 | Try a number bigger than 5/3 like 10.
5 - 30 is gonna give you a negative number so it can't be greater than 5/3. Now you've got the denominator. The denominator cannot be 0 so x ≠ -6. And that's your answer. And if you wanted to do it in interval notation it's a little annoying. But let's see we would start at 5/3 and go down to -6 |
| 42:03 | And from -6 you'd go to negative infinity.
Like that, or to make it more legible It would look like that, okay? How did we like that problem? That was sort of hard because it hard all the negatives sort of backwards and stuff. |
| 42:30 | Were you able to do it?
Look at that. So everybody have a nice weekend. |