WEBVTT
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So I said that the hardest part of calculus is
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algebra. The other hardest part is probably notation and
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just getting a real sense of going from okay,
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just solving equations to really understanding how to communicate in
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the math language. And so we're going to do
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just a very brief introduction to set notation because we're
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gonna move into talking about functions defined on sets.
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And so the first thing we have is the natural
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numbers. You'll see this little blackboard in is the
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symbol. So that's just the counting numbers. 1234
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etcetera. You have the imagers that's denoted by the
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blackboard, see, And we include zero and all
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of the negative numbers. Then we have the rational
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numbers denoted with blackboard que and these air basically fractions
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. But they're just questions of integers where, of
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course, the denominator cannot be zero. We can
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divide by zero, and then what we're really interested
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in is the real numbers. That's what calculus is
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really going to focus on, is gonna focus on
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the study of functions defined on the real numbers.
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And so what are the real numbers? Well,
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it's pretty easy to follow how to define the rational
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numbers. If you believe, uh, there these
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things called the counting numbers that allow us to enumerates
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sets and then the integers you just sort of generalized
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you had in zero, you know, which was
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a big deal. You had a negative numbers,
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which was also a big deal. And then you
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just take questions of those to get the rational numbers
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. But what are the real numbers? Well,
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the real numbers are just the rational numbers with all
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of the holes filled in. And I'm not going
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to spend a lot of time saying much about that
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. If you take a class in advanced calculus or
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analysis is what it's called, then you'll talk a
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lot more about how the real numbers are basically defined
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from the rational numbers. And so I guess the
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only thing I'll say is, What do I mean
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here by hole? Well, it turns out that
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there holtz in the rational numbers. So in other
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words, if I think about well, is there
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a real number X such that when I square it
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So I multiplied by itself? I get to Well
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, it turns out that X is not rational,
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so if I define this thing called Let's just say
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the positive square to two. The number that when
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I squared I get to this X is not going
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to be a rational number, which is kind of
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a deep problem. You know, somehow you might
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think that all numbers irrational, but it turns out
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that their numbers that are not rational and so those
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air kind of the holes that exist within the rational
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numbers and and that's really all I want to say
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about that. But what I do want to connect
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up is, once we have this idea of the
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real numbers, calculus is pretty easy to nail down
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. So calculus is study of functions defined on the
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real line or, more specifically, on subsets of
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the rial line. What we'll do next is will
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actually be a little bit more specific about what types
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of subsets we're going to focus in on. So
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when we get into different types of functions, we're
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going to start throwing out domains of those functions,
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and we're going to get a lot of things that
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look like this. Closed intervals, open intervals.
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This is kind of halfway a joke club and intervals
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so closed and open intervals closed, raise open,
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raise. And then we can combine intervals with these
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operations unions and intersections, and I'll just leave This
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here is a reference you can look it over and
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try to make the connections with everything. I hope
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that you've seen this before, but we just want
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to review the main idea. So notice, You
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know, first of all, notation Aly that when
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we use a square bracket, we're including that in
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point. When we use parentheses, we're not Including
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that in point. Uh, on the right hand
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side here, I'm using this set builder notation.
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So we'll see this time and time again, especially
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if you go into any more advanced math classes.
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It comes up a lot more, but the idea
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here is that if I write okay, so I'm
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saying I want a real number x two and X
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. That's in the real numbers such that some property
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of X is true. So you see, here
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the property is that X lies between these two numbers
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A and B, but in general it's just saying
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I want a real number, such that some property
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, that real number is true and then that defines
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us that now the set might be empty. There
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may be no numbers X that satisfy this property,
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but that's the idea behind set builder notation. And
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we'll see it here as so So the last thing
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I'll mention is that actually weaken, right. The
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entire real line in interval notation is just minus infinity
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to infinity. And, you know, there's also
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these cool diagrams you can draw. So, for
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instance, if I have this interval here, then
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Aiken plot this on the number line like so So
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here in zero here would be one. And so
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I can plotted by putting a solid 0.0 a solid
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dot at one and connecting them to show that Okay
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, this interval 01 is plotted like that. Okay
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, there's similar waste. A plot, all these
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other types of of intervals. For instance, if
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I wanted to plot this ray to do infinity where
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I don't include two Well, then what I could
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do because I started to and I draw a open
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circle to signify that. Okay, I'm not including
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two. This is too. And then I can
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just extend off with an arrow this way, showing
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you know im, including every real number greater than
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to, but not including two. So we use
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this idea again, and we'll actually generalize that idea
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later. When we look at Functions way, start
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drawing the graphs of functions in the X Y plane
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. So the next thing I want to dio is
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introduced a very important operation, and later we'll see
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it's actually a function, but that's the absolute value
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of a real number. So have a real number
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X. The absolute value is defined like this.
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It's defined, actually is a piece wise function,
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so it's just going to be itself. So in
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other words, the absolute value of the rial number
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will be itself if that number is greater than or
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equal to zero. But then, if it's less
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than zero or negative, I'm just going to say
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that the absolute value of a number is negative.
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So if it is negative, it will become positive
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. And this is also a great chance to introduce
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the piece wise notation that I'm sure you're familiar with
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. So for different parts of the domain of dysfunction
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or for different real numbers, it has a different
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definition, and we're going to see this time and
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time again. So again, lock this in your
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mind because it's really important. But notice that there's
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another way to define the absolute value of a number
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, and that's to take the square root of the
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number square and notice what happens here. If a
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number is positive and I square it and then square
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root it, I just get the number back.
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But if the numbers negative, I square it,
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I get it's positive square, and when I take
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the positive square root I get, it's absolute value
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. All right, cool. So this is going
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to come up a lot of the absolute value when
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solving equations and inequalities. So the thing to consider
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is that if I have a variable X, and
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I know that the absolute value of X is equal
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to some real number, a, then one of
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two things could be the case. Either excess a
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or minus X is equal to a because I have
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to consider both parts of the definition. Either ex
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congee A or the negative of X can be a
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and so that's what this equation means. One of
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these two things can be true to make the opposite
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value of X equal today. All right, So
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now that we're getting a handle on the idea of
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sets and subsets specifically of the rial line, let's
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just go ahead and reintroduce the idea of the function
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. And this is going to seem a little bit
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abstract at first. But it's really important to kind
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of get this idea in your head. So a
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function which will note for now just by a little
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F Sometimes we use different letters, but a function
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f from a set de to set why and now
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these sets D and why we really want to think
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about them is subsets of the real line. So
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maybe an interval, maybe a union of two intervals
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, something like that. And so to give the
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definition, I'm just going to draw a picture.
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So we're going to start with our set D,
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and I'm just going to draw it as a well
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, not really a circle but region. So here's
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D, and here's why. And so what it
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does is it takes something in D. So let's
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say X is something in D, and what it's
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going to do is it's going to take it over
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into something in walking and we call this something And
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why? Well, little why like that? Okay
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, and this is the function. So the function
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is going to take everything in this set D in
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a sign, each thing indeed to exactly one thing
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and why and now that's really important. That's if
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you recall. That's like passing the vertical line test
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. So for every input of the function, we
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get exactly one output. And now the said D
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is called the domain of F. In this set
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, why is sometimes called the co domain? That
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where it isn't as Communists domain? That's what it's
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called. And so what we all right is that
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this element why in this in the code of Maine
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is a function, or is an output of this
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input X in the domain? And when we use
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why an X for these variables will say that X
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is the independent variable for the function. And why
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is the dependent variable? Because it depends on which
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X we choose for an input, but we're sort
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of free to independently choose X from the demand.
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But then we once we picks ex, were sort
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of fixed into which output we get in white.
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So that's the basic idea of functions. And just
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to kind of refresh your memory will next give some
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examples of why functions or useful. So here,
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just a couple of examples are actually specifically three examples
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of different types of functions you may encounter. And
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so let's just identify what the independent and the dependent
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variables are. So in the first example, the
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position of a ball, which is given by the
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dependent variable s is a function of the independent variable
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t. So for each time after the ball was
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kicked, I plug in t into my input and
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I get my output, which is a position of
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the ball. Same thing here. The deepening variable
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here is H the height of the plant, and
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it depends on the independent variable D, which is
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the days after it was planning. So I have
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input output. Same thing here, Deepen it.
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Variable is pressure, Pete, and it depends on
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the independent variable T, which is the temperature outside
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. So I plug in temperature outside, and that
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should give me ah, unique pressure in my car's
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tires. So there's a few examples, and I'm
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sure you can think of a lot of examples on
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your own. So I've talked about the domain and
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the co domain of a function. But in the
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calculus setting and in a lot of settings, we're
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not explicitly given. A Domain and ako domain for
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a function were just given, ah, function were
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given that Why is some function of X? Maybe
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this is X squared sine of X. Either the
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X. We're going to talk a lot about those
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examples later. But let's just say for now we
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have a function and we have a dependent variable.
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Why independent variable X. We're going to define two
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very important sense in the first one is called the
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natural domain of F, and now notice. This
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is a little bit different than the domain of that
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, because I'm not actually going to specify what the
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domain is. I'm calling it the natural domain because
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it's exactly that. It's basically all of the rial
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numbers X, for which ffx is a real number
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. So let's actually write that in set builder notation
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so well denote it by D of F. And
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what this is going to be is it's going to
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be all the real numbers such that ffx is a
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real number. So think about, for instance,
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if the function was one over X, well,
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we can't divide by zero so X would be excluded
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from the domain of one over X because 1/0 is
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not a real number. So we're just looking at
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the numbers, the real numbers that we can naturally
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plug in to the function itself. The next concept
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is one that I'm sure you're familiar with. And
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that's the range of the function f. And so
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the range of F, which will denote are a
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Beth is gonna be the set of real numbers.
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Why, through which, what? Well, for
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which there is and X in the domain of f
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such that why is equal to ethics. So in
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other words, we're just looking at all of the
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real numbers that air out, put it by the
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function. So you you look at all possible inputs
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and then you just see what happens when I plug
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in those inputs into my function. The collection of
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all possible outputs is the range of F. So
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here we have a bunch of different ways that we
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can combine functions. And again, I'm sure a
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lot of these air familiar to you. But I
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just want to put him here just as a reference
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and just to review and notice that I've also included
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for each one of these ways to combine functions,
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the definition and also what the domain is going to
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be. So, for instance, if I have
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a function F, I can take a constant multiple
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of that function. So you just multiply the function
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by that constant multiple, and that's just gonna multiply
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all the outputs. By that, whatever that real
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number is C and that's not going to change the
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domain of F. And then we can add two
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functions. We'll see that the domain is just going
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to be the intersection of the two functions. The
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intersection of the domain of the two functions product of
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two functions, is the same thing. You just
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take the product of the outputs and the domain is
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like I said, it's the same. It's the
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intersection. Question is just like product. But we
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just need to add this extra restriction that the function
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that sends a denominator can never be zero And then
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finally the composition of functions is just going to be
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00:19:52.220 --> 00:19:55.599 A:middle L:90%
well. You take the input of one function as
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the output of the other, and the domain is
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going to be basically wherever that makes sense. Eso
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Here it is in and set notation. You could
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look at that, but basically you just want okay
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, I want the output of G to be in
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00:20:10.829 --> 00:20:12.250 A:middle L:90%
the domain of F. Otherwise it's not going toe
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make sense to compose the functions. Another thing to
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00:20:15.720 --> 00:20:19.970 A:middle L:90%
notice about composition of functions is that it is in
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general, not community, meaning that if I take
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f of g of X, that in general is
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not going to be the same thing as G of
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F FX, and that's gonna be important later on
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. We're going to see some examples that really throw
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00:20:38.859 --> 00:20:41.279 A:middle L:90%
some curve balls and maybe the way you think about
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functions. So I just mentioned that function composition in
307
00:20:53.910 --> 00:20:57.279 A:middle L:90%
general is not community. So, for instance,
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let's suppose that ffx is equal to X squared and
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g of X is equal to square root of X
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00:21:11.839 --> 00:21:14.869 A:middle L:90%
. Okay, so let's think about what is f
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00:21:15.640 --> 00:21:21.200 A:middle L:90%
of g of X, well, G just tells
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me to take the square root of the real number
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, and F tells me to square it. So
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what this is going to be is it's going to
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be the square root of X squared, which is
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X with one condition. Notice that we first have
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to take the square root of the number. So
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that means the domain of this composition is going to
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be the same is the domain of Jeep, meaning
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that the domain is not going to include any negative
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numbers. So the domain here is going to be
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00:22:00.539 --> 00:22:03.490 A:middle L:90%
X greater than or equal to zero. So this
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00:22:03.490 --> 00:22:07.200 A:middle L:90%
isn't really the function X where I could just plug
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00:22:07.200 --> 00:22:10.359 A:middle L:90%
anything yet. It has a natural domain restriction.
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So what about G off athletics? Well, this
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00:22:17.789 --> 00:22:22.059 A:middle L:90%
is going to be the square root of X squared
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. And now what is this? This is something
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00:22:23.759 --> 00:22:26.319 A:middle L:90%
that we've already seen. This is absolute value of
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00:22:26.319 --> 00:22:30.099 A:middle L:90%
X. Now, the domain of this function is
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00:22:30.099 --> 00:22:33.029 A:middle L:90%
going to be all real numbers. But actually,
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00:22:33.029 --> 00:22:34.920 A:middle L:90%
the output or the range is what's interesting. We're
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00:22:34.920 --> 00:22:37.890 A:middle L:90%
not going to get any negative numbers. We're only
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00:22:37.890 --> 00:22:41.880 A:middle L:90%
going to get non negative numbers. And so you
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00:22:41.880 --> 00:22:44.670 A:middle L:90%
see, that these two functions are different. They
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00:22:44.670 --> 00:22:48.450 A:middle L:90%
have different natural domains because I can't plug in a
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00:22:48.450 --> 00:22:51.869 A:middle L:90%
negative number here, and I can plug in a
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00:22:51.869 --> 00:22:53.619 A:middle L:90%
negative number here. So you see that the function
338
00:22:53.619 --> 00:22:56.819 A:middle L:90%
composition is not the same, and this is an
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00:22:56.819 --> 00:23:00.680 A:middle L:90%
extremely important example. And I would really encourage you
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00:23:02.440 --> 00:23:04.289 A:middle L:90%
if you're saying what is going on right now to
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00:23:04.289 --> 00:23:08.700 A:middle L:90%
really sit down and think about this example, because
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00:23:08.700 --> 00:23:14.549 A:middle L:90%
it illustrates how intricate functions can be. And when
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00:23:14.549 --> 00:23:17.130 A:middle L:90%
you start composing functions, how careful you need to
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00:23:17.130 --> 00:23:19.940 A:middle L:90%
be later on, we're going to see taking the
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00:23:19.940 --> 00:23:22.319 A:middle L:90%
square root of something squared. We're gonna have to
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00:23:22.319 --> 00:23:26.400 A:middle L:90%
think about that as being the absolute value. Andi
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00:23:26.410 --> 00:23:29.940 A:middle L:90%
and it's really interesting and it's really worth spending some
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00:23:29.940 --> 00:23:34.529 A:middle L:90%
time thinking about. But we actually want to think
349
00:23:34.529 --> 00:23:41.000 A:middle L:90%
about the case where dysfunction composition is community. It
350
00:23:41.000 --> 00:23:44.329 A:middle L:90%
does go both ways, and actually in the special
351
00:23:44.329 --> 00:23:53.160 A:middle L:90%
case where F of G A backs equals G of
352
00:23:53.160 --> 00:24:02.869 A:middle L:90%
Alfa Becks equals X, then we actually give these
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00:24:02.869 --> 00:24:08.410 A:middle L:90%
functions a special name. F N G are said
354
00:24:08.410 --> 00:24:21.420 A:middle L:90%
to be in verses and a lot of times will
355
00:24:21.420 --> 00:24:26.539 A:middle L:90%
write the inverse of F in a special way.
356
00:24:26.549 --> 00:24:36.180 A:middle L:90%
So the inverse of a function f is denoted Yeah
357
00:24:36.740 --> 00:24:41.819 A:middle L:90%
, to the negative one that members. And so
358
00:24:41.819 --> 00:24:44.849 A:middle L:90%
another thing to note about the domain and range of
359
00:24:44.849 --> 00:24:48.369 A:middle L:90%
inverse functions is this. Well, the domain of
360
00:24:48.380 --> 00:24:55.539 A:middle L:90%
F is gonna be the range of f inverse.
361
00:24:55.720 --> 00:25:00.460 A:middle L:90%
That's really nice to know in the range of F
362
00:25:00.740 --> 00:25:03.779 A:middle L:90%
is going to be the domain of F members.
363
00:25:03.440 --> 00:25:06.269 A:middle L:90%
And this really makes sense if you think about it
364
00:25:06.740 --> 00:25:08.569 A:middle L:90%
, because to be in vertebral, if I start
365
00:25:08.579 --> 00:25:11.779 A:middle L:90%
in the domain of F and I apply f I
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00:25:11.779 --> 00:25:14.759 A:middle L:90%
end up in the range of death, Well,
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00:25:14.759 --> 00:25:15.630 A:middle L:90%
that better be in the domain of Beth members,
368
00:25:15.630 --> 00:25:19.029 A:middle L:90%
because I need to be able to go back into
369
00:25:19.029 --> 00:25:22.599 A:middle L:90%
the domain of death to sort of reverse the process
370
00:25:22.440 --> 00:25:26.109 A:middle L:90%
. So again, function in versus is something that's
371
00:25:26.109 --> 00:25:29.349 A:middle L:90%
gonna play an important role and calculus. And so
372
00:25:30.039 --> 00:25:33.019 A:middle L:90%
it's gonna be really important to understand. Maybe the
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00:25:33.029 --> 00:25:37.609 A:middle L:90%
simplest example of two functions that do not have inverse
374
00:25:37.609 --> 00:25:44.569 A:middle L:90%
is X squared and square root of X. So
375
00:25:44.569 --> 00:25:47.490 A:middle L:90%
this leads us to a really natural question. And
376
00:25:47.490 --> 00:25:57.680 A:middle L:90%
the question is, this does every function have an
377
00:25:57.680 --> 00:26:07.009 A:middle L:90%
inverse? And the answer, of course, is
378
00:26:07.009 --> 00:26:10.839 A:middle L:90%
no. And we've actually already seen an example of
379
00:26:10.839 --> 00:26:14.140 A:middle L:90%
a function that does not have an inverse on its
380
00:26:14.140 --> 00:26:18.559 A:middle L:90%
natural domain. And that function is X squared.
381
00:26:21.140 --> 00:26:23.549 A:middle L:90%
And the reason it doesn't have an inverse is because
382
00:26:23.710 --> 00:26:26.549 A:middle L:90%
, let's say I started with the number two and
383
00:26:26.549 --> 00:26:30.529 A:middle L:90%
I squared it when I went to go in square
384
00:26:30.529 --> 00:26:33.250 A:middle L:90%
root or reverse the process. I have to make
385
00:26:33.250 --> 00:26:36.210 A:middle L:90%
a choice. Do I mean to or do I
386
00:26:36.210 --> 00:26:38.259 A:middle L:90%
mean minus two? There are actually two square roots
387
00:26:38.529 --> 00:26:41.650 A:middle L:90%
of four, and so, in a sense,
388
00:26:41.660 --> 00:26:45.849 A:middle L:90%
you cannot put the toothpaste back in the two for
389
00:26:45.849 --> 00:26:48.410 A:middle L:90%
every function. So then the next natural question to
390
00:26:48.410 --> 00:26:52.789 A:middle L:90%
ask is, When does the function having members?
391
00:27:04.940 --> 00:27:08.009 A:middle L:90%
And if you recall to be a function for every
392
00:27:08.019 --> 00:27:11.160 A:middle L:90%
input, you had to have exactly one output.
393
00:27:11.940 --> 00:27:15.470 A:middle L:90%
So if we want a chance to reverse the process
394
00:27:15.480 --> 00:27:19.680 A:middle L:90%
, we better have for each output exactly one input
395
00:27:21.539 --> 00:27:22.559 A:middle L:90%
. And so, in other words, we want
396
00:27:22.559 --> 00:27:26.289 A:middle L:90%
the function to be one toe one. So the
397
00:27:26.289 --> 00:27:40.420 A:middle L:90%
answer is F R function has an inverse if it
398
00:27:40.420 --> 00:27:48.000 A:middle L:90%
is okay once one. And so what does 11
399
00:27:48.000 --> 00:27:51.609 A:middle L:90%
mean? Well, it's exactly what I just said
400
00:27:51.619 --> 00:27:55.210 A:middle L:90%
for each output. There's exactly one input, and
401
00:27:55.210 --> 00:27:57.759 A:middle L:90%
we can write this symbolically. So suppose that I
402
00:27:57.759 --> 00:28:02.839 A:middle L:90%
had f of X equaling f of Z. So
403
00:28:02.839 --> 00:28:06.529 A:middle L:90%
I had to numbers X and Z that gave me
404
00:28:06.529 --> 00:28:08.430 A:middle L:90%
the same out. But well, then it better
405
00:28:08.430 --> 00:28:14.819 A:middle L:90%
be the case. If this happens, that X
406
00:28:14.829 --> 00:28:17.970 A:middle L:90%
is actually just equal to Z. So it's not
407
00:28:17.970 --> 00:28:21.279 A:middle L:90%
possible. Toe have two different inputs giving me the
408
00:28:21.279 --> 00:28:23.410 A:middle L:90%
same output. And you can see that when this
409
00:28:23.410 --> 00:28:26.619 A:middle L:90%
is the case, the inverse function is actually well
410
00:28:26.619 --> 00:28:30.049 A:middle L:90%
defined. It actually is itself a function because,
411
00:28:30.049 --> 00:28:33.250 A:middle L:90%
remember, my outputs become my inputs for an inverse
412
00:28:33.250 --> 00:28:37.690 A:middle L:90%
function. And so if my output is here,
413
00:28:37.859 --> 00:28:41.099 A:middle L:90%
I better have one input back in the domain of
414
00:28:41.099 --> 00:28:51.430 A:middle L:90%
the original function. So now that we know when
415
00:28:51.430 --> 00:28:53.710 A:middle L:90%
a function has an inverse, let's actually go back
416
00:28:53.940 --> 00:28:56.710 A:middle L:90%
to our example of a function that does not have
417
00:28:56.710 --> 00:29:00.789 A:middle L:90%
an inverse on its natural domain. And that's X
418
00:29:00.789 --> 00:29:10.900 A:middle L:90%
squared has no in verse and another way to say
419
00:29:10.900 --> 00:29:12.390 A:middle L:90%
this that will say sometimes so it doesn't have a
420
00:29:12.390 --> 00:29:17.460 A:middle L:90%
members. In other words, it's not in veritable
421
00:29:18.539 --> 00:29:22.049 A:middle L:90%
and specifically it's not in vertebral on its natural domain
422
00:29:23.039 --> 00:29:27.059 A:middle L:90%
. But what would happen if I just restricted the
423
00:29:27.059 --> 00:29:30.430 A:middle L:90%
domain. So I know that, too, and
424
00:29:30.430 --> 00:29:33.380 A:middle L:90%
negative two are going to give me the same output
425
00:29:33.390 --> 00:29:34.660 A:middle L:90%
when I square them. But what if I only
426
00:29:34.660 --> 00:29:37.839 A:middle L:90%
consider the non negative numbers? So if I put
427
00:29:37.839 --> 00:29:44.839 A:middle L:90%
a domain restriction and by domain restriction, all I'm
428
00:29:44.839 --> 00:29:48.400 A:middle L:90%
doing is just throwing out certain values that I don't
429
00:29:48.400 --> 00:29:51.329 A:middle L:90%
want to consider in the domain and in this case
430
00:29:51.700 --> 00:29:53.200 A:middle L:90%
, those air the negative numbers. So I'm going
431
00:29:53.200 --> 00:29:59.119 A:middle L:90%
to restrict my function F to the set 02 Infinity
432
00:29:59.130 --> 00:30:04.049 A:middle L:90%
and I'll include zero. And so on this restriction
433
00:30:11.740 --> 00:30:17.259 A:middle L:90%
, the function X squared is one the one.
434
00:30:22.740 --> 00:30:32.559 A:middle L:90%
Thus it's in vertical, but it's on Lian vertebral
435
00:30:32.559 --> 00:30:37.099 A:middle L:90%
when I consider this function X squared on this domain
436
00:30:37.109 --> 00:30:41.519 A:middle L:90%
restriction the ray from zero to infinity. And of
437
00:30:41.519 --> 00:30:45.660 A:middle L:90%
course, in this case, the inverse is exactly
438
00:30:45.660 --> 00:30:48.480 A:middle L:90%
what we expected to be just square root of X
439
00:30:51.940 --> 00:30:53.440 A:middle L:90%
. So this brings up a very subtle point that
440
00:30:53.440 --> 00:30:56.750 A:middle L:90%
we'll see again. We have a function that is
441
00:30:56.750 --> 00:31:00.119 A:middle L:90%
not one toe one, so it's not convertible on
442
00:31:00.119 --> 00:31:03.390 A:middle L:90%
its natural domain. However, if we restrict the
443
00:31:03.390 --> 00:31:07.529 A:middle L:90%
domain to a smaller set where the function is 1
444
00:31:07.529 --> 00:31:11.420 A:middle L:90%
to 1 than it is in veritable, and we
445
00:31:11.420 --> 00:31:14.140 A:middle L:90%
confined the numbers. We're going to see this again
446
00:31:14.140 --> 00:31:15.980 A:middle L:90%
when we talk about the trig and metric functions in
447
00:31:15.980 --> 00:31:18.400 A:middle L:90%
general, Trig and metric functions will not be one
448
00:31:18.400 --> 00:31:21.869 A:middle L:90%
toe one they're not gonna have in verses. So
449
00:31:21.869 --> 00:31:25.859 A:middle L:90%
we're gonna have to narrow in on a small interval
450
00:31:25.859 --> 00:31:27.440 A:middle L:90%
for which they are 1 to 1, and then
451
00:31:27.440 --> 00:31:30.960 A:middle L:90%
we'll be able to define it embers, So it's
452
00:31:30.960 --> 00:31:33.680 A:middle L:90%
a little tricky. But again, if you can
453
00:31:33.680 --> 00:31:37.750 A:middle L:90%
understand this example of x squared and squared of X
454
00:31:37.759 --> 00:31:40.920 A:middle L:90%
and realize that their Onley in verses, when I
455
00:31:40.920 --> 00:31:44.390 A:middle L:90%
restrict the domain, then you're gonna be taking a
456
00:31:44.390 --> 00:31:48.230 A:middle L:90%
lot of huge steps towards understanding kind of the basic
457
00:31:48.230 --> 00:31:51.950 A:middle L:90%
ideas behind functions that we need to understand moving forward
458
--> A:middle L:90%
.