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Robert McKeown: Hello and welcome to ALEKS
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walkthrough video number five. My name is Robert
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McKeown and today we're going to be talking
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about functions and graphs, students. In my
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experience at York University, and I'm not been
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here very long, students find this topic
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extremely challenging. I don't think it's
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particularly hard. But what you have to do is
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you have to memorize a number of definitions,
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and then apply those definitions to different
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situations. And that can be a little bit
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difficult because you actually have to learn the
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definitions, you actually have to do the work
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doesn't matter how smart you are smart is not
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going to help you if you don't have the
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knowledge necessary to answer the question. And
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so this is actually the longest series of videos
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of all a topics because there's just so much
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that I want to explain to you. And then you just
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need to know. So you can think of maybe the
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first four topics as being rather introductory,
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didn't require a lot of background information
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or background knowledge, background knowledge,
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now we're building off the knowledge you learned
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in topics 123, and four, and we're going forward
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into more advanced mathematics and into economic
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applications. So again, before we get started,
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make sure you have a pencil, a pen, paper, or if
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you're really fancy, get yourself a tablet with
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a writing pen. That's a really great tool to
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have. I was so impressed with the my students
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who had one that I went out and bought one. And
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I use it in class. If you've ever had a lecture
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with me, you'll see it for sure. So without any
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further ado, let's get down to the problems. To
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answer the topic five questions correctly, you
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have to understand a few definitions. And our
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first definition is what is a function. So you
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can read a formal definition on the slides. One
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thing I'd like you to think about is I'd like
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you to think about a streetlight. So functions
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don't have to be mathematical in nature, we
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could have an X variable, which is the
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streetlight can be a red light, can be red. Or
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it can be a green light.
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And there's a function that transforms that
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observing that light and to an outcome, which is
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stop your car forever the red light, and if it's
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a green light, it's go. And so we've got two
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ordered pairs here, we've got an ordered pair
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that's red light, and associated with stop and
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another ordered pair, that's a green light. And
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it leads to go. So that's an example of a
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function in everyday life. We're gonna focus on
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numerical functions here. But in economics,
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sometimes we don't want to use numbers, we want
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to use concepts like this, and we do so here is
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another definition, maybe a simpler definition
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of what a function is. So we've got a function
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something like this. And we might have y is
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equal to some function of x. So let's consider a
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function x squared. So if x is equal to zero, y
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is going to also be equal to zero. And if x is
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equal to one, y will be equal to one. If x is
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equal to negative one, y is also equal to one.
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And so I'll tell you that x squared is a
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function for every x there is one Y value. Now I
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haven't proved it. But if you want, you can go
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home, you can put in any number you want into
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that. And you're only going to get one y value,
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whatever number you put into that function,
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you're going to get only one y value. In fact,
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that's what makes it a function. Let me show you
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an example of something that is not a function.
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So this is a function. What if I have? I can't
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use that terminology, what if I have y squared
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plus x squared is equal to 25? Well, I've got y
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is equal to a function of x. So why don't I
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rearrange this thing over here and isolate y on
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one side, so that it looks like the function I
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showed you on the left hand side of the screen.
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So I can take, I know that y squared is going to
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be equal to 25 minus x squared, I can write y,
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I'll take the square root of both sides of the
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expression. So I've got 25 minus x squared. But
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notice that this is going to be plus or minus,
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whenever I take the square root of something
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that's been squared, it could be positive or
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negative. And so with this example, for each x
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value, there are two possible I'm not going to
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say each, let's just say for some,
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there are two possible y values. So it's not a
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function, the language we like to use in
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mathematics and economics is that the x values
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are in something called the domain. And the y
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values are in something called the range. So if
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I've got a set of axes here, and maybe I've got
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one, or 01, and two, here, this is the domain.
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And then there's some function
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that takes us to,
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I'm gonna just change this one to a negative
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one, like our previous example. And negative one
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also maps to Y equals one. So the function is
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mapping from the domain into the range. And if
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you want to read more about it, I found a neat
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website down here that you can read more about
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domain and range and what a function is what a
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function is not. There is a special kind of
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function, and it's called a one to one function.
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Now, when we had this previous example, we had a
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function that was equal to x squared. But you
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can see here that, and I'll get rid of that
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because we don't care about it. But you can see
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that there's more than one x value factor or two
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that lead to one y value. So this is not a one
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to one function.
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Now let me see.
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I could write a one to one function,
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which is just f of x is equal to x And notice
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that each number in the domain leads to a unique
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number in the range.
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So f of x equal to x is a one to one function.
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So there's, this is a function over here, but
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it's not a one to one function.
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It's a function but not a one to one function
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over here. This is a function and it is a one to
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one function.
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One to One functions are important because every
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one to one function has an inverse function has
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an inverse function. So let's consider another
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one to one function, I'll have 01 and two in my
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domain. And this time, I'll just say let the
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function be equal to two x. And there's my
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domain. And I'm going to have Y values here, if
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x is zero, y is zero. If x is one, y is two, and
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if x is two,
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if x is two, y is four. Now let's consider the
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inverse. So if we have the inverse of this
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function, this functions unique inverse, then
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it's going to reverse, essentially just going to
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reverse the mapping.
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And the inverse of this function is equal to one
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half times x.
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So inverse functions are not very, very
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complicated, in their fundamental or simplest
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form, they're just reversing. Where we're moving
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from, are we moving from X to get y or are we
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moving? Are we taking y to get X inverse
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functions have a very interesting property. So
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this little operator here creates a composite.
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So this is a to operator to create a composite
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function. And more commonly, it's written like
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this, which is basically to say that we have the
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inverse function and we plug it into the
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original function. We're going to end up with
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just x whatever that x value is. So let me give
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you an example from the previous one, we had f
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of x is equal to two the inverse of that
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function or excuse me, that was two x, the
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inverse was equal to one half x. and say I
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wanted to evaluate
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we always start with the inner right whatever is
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in the brackets is the first order of
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operations. So F inverse three
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We have one half
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times three, which gives us
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three over two. Now let's take that three over
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two and plug it into the original function. And
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we get two times three over two, which is equal
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to three. And that is exactly what this special
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property is all about. And it says that if we
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take the inverse of x, and then we take that
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inverse of x and take the function of the whole
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thing, we get x. We've gone through an
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introduction together, let's apply what we've
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learned to answer the next three questions. So
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we're given two functions. The first function g
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doesn't have any calculations, it's just a set
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of ordered pairs. So given us the ordered pairs
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directly, we don't have to do any calculation to
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find them. The second function age is a function
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as you're used to seeing it, where we're going
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to have to perform some calculations. Now the
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first question is asking us to find the inverse
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of G. So remember that if we have x, mapping
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into y, according to G, now we've got the
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inverse of G where we're mapping from y to x. So
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g inverse is going to take the y value here,
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move it to the first position where the domain
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is. And if y is equal to seven, we're going to
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map to x equal to negative five. And we're going
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to do that for each of the ordered pairs.
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And the squirrely back, it represents a set
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beginning and the ending of the set. And it's a
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set of ordered pairs. And we're done. So finding
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the inverse function, when you're given a set of
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ordered pairs, is very straightforward, you just
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switch the positions of the domain and the
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range. So we're asked to find the inverse of h
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of x. So I'm going to start it off by writing y
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is equal to three x plus 13. And now I'm going
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to solve for x. So I'm going to isolate x on the
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left hand side of the equation. So I'm going to
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have three x is equal to y minus 13. And then x
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is equal to y minus 13 divided by three, then
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I'm going to replace x with y, and y with x. So
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I'm going to have y is equal to x minus 13 over
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three, so the inverse
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is equal to x minus 13 over three, so that's the
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inverse of the function. So it started switching
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the x's and y's around. But that's just because
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we want to show the inverse function should be a
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function of x. Our next question is asking for
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the composite function h composite with its
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inverse. And it wants to wants us to evaluate
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that composite function when x is equal to four.
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So why don't I do this the hard way, and then we
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can I can show you and remind you how to do it
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the easy way. So another way of writing In this
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is to say that we have H
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inverse of four, and then that that inverse is
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actually within the original function. I've
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taken the inverse function we came up with on
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the previous slide, and brought it with me. And
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so why don't we do that one first. So H, one of
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four is going to be equal to four minus 13
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divided by three, which is equal to negative
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nine over three, which is equal to negative
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three. Now I'm going to use, I'm going to plug
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that negative three into the original function
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through three times negative three, plus 13 is
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equal to negative nine plus 13, which is equal
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to four. Right? And that's exactly what we
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should get because we know that the composite of
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the H function and its inverse I'll delete that
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little x when it's evaluated at four whether
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it's x is going to be equal to four. Right? Just
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like if we didn't have a number to evaluated at,
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we would get our x value