WEBVTT
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Let’s take a look at the term function.
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In everyday language, the word function could be used like this:
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you could say the income that a person earns as an adult
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is a function of the level of education that person attains.
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So what does that mean? That means that a person attains a level of education
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and then from that results an income,
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and that is fine in everyday language,
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but when you talk about a function in mathematics,
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the definition is much more specific.
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There is one particular criteria that a function must have.
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Here is what it is: for each item in the first group, there must be one
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and there must be only one item that results from that in the second group.
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Alright, let’s take a look at some illustrations from everyday life.
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This first one is an illustration of a situation that is not a function.
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We will take a group of people. Let’s call them Larry, Michael, and Joel.
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These will be the items in the first group.
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The function that is going to produce the items in the second group
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would be to list their phone numbers.
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Okay, so let’s see. Take Larry and we will list his phone number: 642-8729.
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Then we will take Michael, and when we ask him his phone number
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he says: 931-1846 or 766-6956.
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When we ask Joel he says his phone number is: 862-4193.
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Why is this not a function? It is not a function because this one element in the first set
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maps to two elements in the second set,
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and that violates the definition of a function.
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So this situation does not represent a function.
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Alright, let’s look at another illustration.
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We use the same three people,
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but we will use a different function.
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So this time the function will be list the social security numbers.
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Okay so Larry’s social security number is 438-61-2915.
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Michael’s social security number is 434-64-3928.
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Joel’s social security number is 435-19-2003.
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Does this represent a function?
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Yes it does because for each element in this set there is one
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and only one element in this set.
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Let’s take a look at one more example.
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Another everyday situation that you are familiar with
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is the fact that a person has a birth date.
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In this case, we will use month and date. So this is person was born on.
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Okay, so Larry was born on October the 19th.
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Michael was born on January the 24th.
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It turns out that Joel was also born on January the 24th.
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The question is: does this represent a function? Well let’s see.
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For each element in the first set, there is one
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and only one element in the second set.
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It is perfectly okay that these two elements here to
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match to the same element in the second set.
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Yes, this represents a function.
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The elements in this set are called the domain.
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The elements in this set are called the range.
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Let’s take a look at some examples.
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Are these functions? If yes, state the domain.
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The domain is the set of all elements in the first set.
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Let’s look at part a. We have three ordered pairs: (2,4), (2,6), and (2,8).
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Let’s draw this. The 2 in the first ordered pair maps to 4.
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The 2 again in the second ordered pair maps to 6.
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The 2 in the third ordered pair maps to 8.
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What we have here is one element but it maps to three different elements here.
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This violates the criteria for a function that says that each element here
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maps to one and only one element here.
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So no, this does not represent a function.
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In part b, we again have three ordered pairs: (2,1), (3,1), and (5,7).
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The 2 maps to 1, the 3 also maps to 1, and then the 5 maps to 7.
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This is equivalent to two different people having the same birthday,
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and that is perfectly fine. That meets the definition of a function.
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Yes, this is a function. Now because the answer is yes,
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we have to state the domain.
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We are going to state the domain here as a set,
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and it is going to contain three elements: 2, 3, and 5.
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In part c we have a slightly different notation. We have a set defined here.
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This is read: the set of all ordered pairs, (x,y),
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such that y is equal to 2x minus 1.
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So the rule here, or correspondence,
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is take an element in the domain, multiply it by 2,
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subtract 1, and what you get is the resulting element
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in the range, which is y. Alright, so let’s think about this.
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Well in this case, x can be any real number.
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There is no reason why we could not use any number: positive, negative, or zero.
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We put it into here, we multiply by 2, subtract 1, and what happens?
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The result is we get a real number.
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What will not happen is that when we put an x in here,
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we will never get 2 different y’s for that x. So, this is a function. Yes.
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The domain, or the choices of the elements in the domain, are going to be all real numbers.
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If we write this in interval notation, we can write it as negative infinity to infinity.
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In some cases, we may just use this capital R symbol. Okay, now let’s look at part d.
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What happens here from part c to part d, is that we realize that overtime,
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this notation for part c is very cumbersome.
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So what happens is we tend to drop part of it,
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and we just use this description of how you get from x to y. Alright so let’s look.
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Again, there is no reason why any value of x would not work here.
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You put it into here, square it, and multiply by 2.
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Then you multiply it by 7, add it this, and add 9. Follow the rule. What happens? You get a y.
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What happens here is you put in an x, and you get one and only one y.
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So yes, this is a function. The domain again is going to be
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either one of these two notations for all real numbers.
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Alright, let’s take a look at part e. You should be suspicious right away when you
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see a variable in the denominator.
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There is going to be some sort of restriction on the domain. What is it?
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Well x minus 6, what? x minus 6 cannot equal 0.
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So that means that x cannot be 6. We see there is a restriction on the domain.
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But what about this is a function?
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Well let’s see. As long as we do not use 6, we are okay,
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because any real value that we put into here will give us one value of y.
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Again, yes this is a function, but the domain is not going to be all real numbers.
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It is going to be either this notation, or depending on the situation,
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you may be able to just put x is a real number not equal to 6.
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It depends on what the directions are and
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what you are trying to do with this. Alright, let’s look at a few more.
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y squared equals x minus 4. Is this a function?
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Well let’s see. Let’s pick a value for x,
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and let's see what happens, and what y we get.
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I am going to pick something for x that gives
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this right hand side a result of a perfect square. So I am going to pick x is 13.
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So what happens is, then I get y squared equals 13 minus 4, which is 9,
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and which means I get y is plus or minus 3.
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Now if you go back and look at what we did with mapping,
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that means that this x value of 13 maps to 3, but it also maps to negative 3.
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What do you know about that?
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You know that what you have here is an element in the domain
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that maps to two different elements in the range,
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and this violates the definition of a function.
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Therefore this does not represent a function.
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Now let’s just stop a minute and think about why this is the case.
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This is the case because when we put a value here, in here for x,
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we get one value here. But in order to solve for y, we get two values.
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So what does that tell you? That tells you that anytime you have a situation
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where x in the independent variable, y is the dependent variable,
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you are going to get a situation when you have y squared,
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where this is not a function. This is because every time you do this and
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you take the square root of both sides, you are going to get two values for y
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that go with the same x. You can start looking for this
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and anytime you see this situation, you will know it is not a function.
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Okay that brings us right into the next one.
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We have x squared plus y squared is 9.
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So what do you know? What did you learn from f?
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You learned that in this case when you substitute a value in for x,
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the resulting values for y, each time you put in one x, you will get two values of y.
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What happens is you know right away that no, this is not a function.
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Okay, in part h we have a restriction on the domain.
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2x minus 1. What is the restriction? Think about it.
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2x minus 1 cannot be negative, which means it has to be what?
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It has to be greater than or equal to 0.
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Which means that 2x has to be greater than or equal to 1.
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Which means x has to be greater than or equal to 1/2.
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So let’s see. Let’s pick a value in this domain.
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Okay so let’s pick something like, oh I don’t know, 6.
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Okay so when x is 6, 6 times 2 is 12, minus 1 is 11.
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We get the square root of 11. We get one value.
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So for this one x we get one y, and this is fine.
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Yes a function. And what is the domain?
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You could write the domain like this,
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or you could write it in interval notation.
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You want to use a closed interval here and infinity here.
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This is the domain of this function. Alright, one last one.
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This expression here has not only a variable under the radical,
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but the radical is in the denominator.
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So let’s look at the domain to start off with.
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Well first of all, x plus 3, it cannot be 0, and it also cannot be negative.
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So what does that leave? Well that means that x plus 3 must be positive,
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so x must be greater than negative 3.
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Okay so let’s pick something greater than negative 3 like say 10.
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10 plus 3 is 13, so you get 4 divided by the square root of 13,
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which is one value of y. So for each x,
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you are going to get one and only one y.
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So yes this is a function. Here is the domain:
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x is greater than negative 3.
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Notice here that this is an open interval,
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not including the negative 3, and so here is the domain.
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Alright, let’s look at a few more things about functions.
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Let’s talk for a second about function notation.
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Now if x represents an element in the domain
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and y represents an element in the range, then what you have is
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that the y you get is simply a function of the x that you choose.
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Okay, now let’s say that again. The y that you get is a function of the x that you choose.
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So y is a function of x. Okay so say that to yourself a few times.
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y is a function of x. Then think about it over and over and over again.
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Just like everything else, we come up with a shortcut.
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What does it look like? It looks like this.
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y is a function of the x that you choose.
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Okay, and so this becomes the standard notation used to represent a function.
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So that means that in an equation, in place of y,
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when you know that this equation represents a function,
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you can simply replace this with f of x.
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It is very useful notation. Let me show you what I mean.
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In example two, the directions are: for the function defined by, and let’s put this
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in a box so it is easy to spot.
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Here is the function we are going to work with.
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This is a function, and we know it is a function
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because right here instead of y, it says f of x.
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Okay so here is the function, we want to find the following:
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f of negative 1. Well this means that x is going to be
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replaced with negative 1.
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So everywhere in this expression, I replace x with negative 1.
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So we have 2 times negative 1 squared, minus negative 1, plus 3.
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Then I am going to evaluate the expression.
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Be careful here with signs. Negative 1 squared is 1. So this would be 2,
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then this would be plus 1, and plus 3, which is 6.
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So what we have is that f of negative 1 is 6. So what does that mean?
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That means when x is negative 1, the resulting y is 6.
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Alright, now this time it says replace all the x’s with the opposite of x.
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So we would have 2 times the opposite of x squared,
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minus and in place of x I am going to put the opposite of x,
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and then plus 3. Okay, the opposite of x squared is x squared.
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So I get for the first term 2x squared.
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Then here I get plus x, and then plus 3.
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So what that means is if I put in the opposite of a value of x,
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this is the resulting expression that I would get.
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Alright now, part c is different. Part c says I am going to
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take the opposite of the whole thing.
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So that means what you are going to do is
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you are going to take the entire right hand side here,
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and replace it here. We would get
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the opposite of then the entire expression.
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We will clean it up a little bit, distribute this negative.
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We will get negative 2x squared, plus x, and then minus 3.
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Okay, part d says now, in place of the original x’s,
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I want you to put the binomial x plus h.
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Alright, so I still have 2, then I have x plus x squared,
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minus, and then I need parenthesis, x plus h, and then plus 3.
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Okay, so I have replaced the original x
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with x plus h here and x plus h here.
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Now I could simplify this, but what I am going to do
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is I will just hold that until the next example.
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In part e, we want to find f of x plus h, as we did in part d,
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then we want to subtract the entire function, f of x,
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and then divide by h. Alright, let’s do this one step at a time.
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First of all, let’s find f of x plus h again.
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This means put x plus h in for this x and this x.
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So I have 2 times, x plus h quantity squared,
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minus x plus h, plus 3. Then that takes care of this part right here.
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Now I need to subtract the function.
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So I am going to subtract 2x squared minus x plus 3.
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Then I am going to take all of that and divide it by h.
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Okay, so that equals, I am going to square this term.
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Now x plus h squared, let’s go up here and do this.
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This would be x plus h squared is x squared plus 2xh plus h squared.
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That is this term. I want to multiply all three of those terms by 2.
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So I would get 2x squared plus 4xh plus 2h squared.
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That takes care of this part. The most common error on this problem
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is right here, is to lose this 2 here
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or to lose this 4 here. So watch out when you do this. Very common error.
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Alright now I am going to distribute
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the negative here, copy over the plus 3,
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and now distribute the negative all the way across.
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Take your time when you do this problem.
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It is really all about signs and handling the algebraic expression,
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and then of course all of this is divided by h.
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Okay, so what happens? Well let’s see.
00:22:54.1500 --> 00:22:58.600
2x squared minus 2x squared is 0.
00:22:59.800 --> 00:23:13.800
Then minus x and plus x is 0, and 3 and minus 3 is 0. If you did it right,
00:23:13.820 --> 00:23:15.900
this is what is going to happen every time.
00:23:15.920 --> 00:23:20.200
All the terms that have no h’s are going to end up canceling,
00:23:20.220 --> 00:23:34.250
and what we have left is 4xh plus 2h squared minus h, all divided by h.
00:23:35.300 --> 00:23:40.300
So what I have is an h as a factor in all three terms in the numerator.
00:23:40.320 --> 00:23:43.550
I am going to factor out the h in the numerator.
00:23:43.570 --> 00:23:53.600
That will leave me 4x, and plus 2h minus 1.
00:23:53.620 --> 00:23:59.850
Again, very common error here is when you factor h out of this term,
00:23:59.870 --> 00:24:03.350
you do not get 0, you get negative 1.
00:24:03.370 --> 00:24:10.200
Then the whole thing is over h, and then now you have h divided by h.
00:24:10.220 --> 00:24:19.700
The final simplification is to get 4x plus 2h minus 1.
00:24:20.900 --> 00:24:24.450
This particular expression turns out to be
00:24:24.470 --> 00:24:26.750
something very important in calculus.
00:24:26.770 --> 00:24:31.100
This is a skill you want to practice and get really good at.
00:24:31.120 --> 00:24:33.150
Let’s look at a few more things.
00:24:36.250 --> 00:24:38.750
The domain of a function is the set of all
00:24:38.770 --> 00:24:45.950
values of x for which the function is defined.
00:24:47.150 --> 00:24:52.350
It is often very helpful to classify a function to determine its domain.
00:24:53.800 --> 00:24:58.700
The first class of functions we will look at is polynomial functions.
00:24:58.720 --> 00:25:06.850
So here is the definition: the function f of x, a sub n x to the n,
00:25:06.870 --> 00:25:10.800
plus a sub n minus 1 x to the n minus 1,
00:25:10.800 --> 00:25:14.450
plus a sub n minus 2 x to the n minus 2,
00:25:14.470 --> 00:25:20.600
plus and it continues in the same manner then, plus a sub 1 x,
00:25:20.620 --> 00:25:26.550
which of course is to the 1, plus a sub 0, which is times x to the 0,
00:25:26.570 --> 00:25:34.600
is a polynomial function of degree n where n is a nonnegative integer,
00:25:34.620 --> 00:25:42.900
and the numbers a sub 0, a sub 1, a sub 2, etc., to a sub n are real numbers.
00:25:42.920 --> 00:25:49.950
The domain of every polynomial function is negative infinity to infinity.
00:25:49.970 --> 00:25:54.850
Now do not be intimidated by all this notation.
00:25:54.870 --> 00:26:00.300
What this simply means is that a sub n is the coefficient that goes with
00:26:00.320 --> 00:26:07.300
x raised to the n power. a sub n minus 1 is the coefficient that goes with
00:26:07.320 --> 00:26:13.7500
x raised to the n minus 1 power, and so forth.
00:26:14.450 --> 00:26:19.450
A second class of functions is rational functions.
00:26:19.470 --> 00:26:26.950
A rational function is a function of the form:
00:26:26.970 --> 00:26:31.850
f of x equals g of x divided by h of x
00:26:32.350 --> 00:26:34.900
where h and g are polynomial functions
00:26:34.920 --> 00:26:38.550
such that h of x does not equal 0.
00:26:38.570 --> 00:26:41.400
So the domain of a rational function is
00:26:41.420 --> 00:26:44.850
the set of all real numbers x such that
00:26:44.870 --> 00:26:47.950
h of x does not equal 0.
00:26:48.750 --> 00:26:51.300
The third class is root functions.
00:26:52.200 --> 00:26:57.750
The function f of x equals the nth root of g of x
00:26:57.770 --> 00:27:05.250
is a root function where n is an integer,
00:27:05.270 --> 00:27:07.450
such that n is greater than or equal to 2.
00:27:07.470 --> 00:27:10.000
Now there are two possibilities for the domain.
00:27:10.020 --> 00:27:18.500
One: if n is even, the domain is then the solution to the inequality
00:27:18.520 --> 00:27:21.200
g of x is greater than or equal to 0.
00:27:22.300 --> 00:27:27.800
The second case is where n is odd, then the domain is
00:27:27.820 --> 00:27:31.900
the set of all real numbers for which g of x is defined.
00:27:31.920 --> 00:27:35.750
Now let’s look at some examples.
00:27:37.650 --> 00:27:43.750
Example three: Classify the given functions as a polynomial function,
00:27:43.770 --> 00:27:48.100
rational function, or root function, and then find the domain.
00:27:48.120 --> 00:27:51.200
Write the domain in interval notation.
00:27:53.350 --> 00:27:58.900
Part a is the function f of x equals 2x minus 11.
00:27:59.550 --> 00:28:02.900
This is a polynomial function. It has degree one.
00:28:02.920 --> 00:28:15.750
The exponent of x here is 1. So we classify it as a polynomial function.
00:28:16.100 --> 00:28:19.150
Because it is a polynomial function,
00:28:19.170 --> 00:28:28.000
the domain, in interval notation, is negative infinity to infinity.
00:28:28.950 --> 00:28:37.850
Alright, let’s look at part b. f of x equals 3x minus 1 divided by 3x plus 1.
00:28:38.700 --> 00:28:44.800
We can see that this function is of the form g of x over h of x,
00:28:44.820 --> 00:28:53.250
and so this is a rational function.
00:28:55.550 --> 00:28:58.200
If there is a restriction on the domain,
00:28:58.220 --> 00:29:08.300
we find it by setting h of x, the denominator, equal to zero.
00:29:08.300 --> 00:29:12.650
What I am going to put here is that h of x does not equal to 0,
00:29:12.670 --> 00:29:23.950
and then I will solve. I subtract 1 from both sides, and divide by 3.
00:29:24.000 --> 00:29:30.900
I see that x is not equal to 1/3. But of course I have to be careful
00:29:30.920 --> 00:29:35.250
because when I subtracted 1 from both sides I should get negative 1 here,
00:29:35.270 --> 00:29:40.000
and negative 1/3. Always check before you go on.
00:29:40.020 --> 00:29:44.450
Now in interval notation, I am going to write this as
00:29:45.800 --> 00:29:56.000
negative infinity to negative 1/3 union negative 1/3 to infinity.
00:29:57.350 --> 00:30:07.950
Let’s look at a few more. Part c: f of x equals the fourth root of 2 minus x.
00:30:07.970 --> 00:30:13.300
We can see that this is a root function where n is 4.
00:30:14.350 --> 00:30:22.800
So let’s classify this as a root function.
00:30:23.900 --> 00:30:33.150
Because 4 is even, the domain will be the values of x
00:30:33.170 --> 00:30:38.950
where 2 minus x is greater than or equal to 0.
00:30:39.000 --> 00:30:47.300
Okay, so we add x to both sides and we get x is less than or equal to 2,
00:30:47.320 --> 00:30:57.600
and in interval notation we get this for the domain.
00:30:59.650 --> 00:31:12.100
Part d: f of x equals the seventh root of 5x plus 9 is also a root function.
00:31:14.000 --> 00:31:19.400
It has the same form but the difference is that here, this is the seventh root.
00:31:19.420 --> 00:31:24.000
So 7 is odd and in that case, without doing any work,
00:31:24.020 --> 00:31:31.500
we know that the domain is negative infinity to infinity.
00:31:31.520 --> 00:31:44.450
Let’s look at one more. Part e: f of x equals the fifth root of x plus 2 divided by x minus 9.
00:31:45.400 --> 00:31:48.350
When you look at this you see the fifth root,
00:31:48.370 --> 00:31:57.350
and so you classify this as a root function.
00:31:59.150 --> 00:32:02.950
Because n is odd, you might think originally
00:32:02.970 --> 00:32:06.200
that the domain would be all real numbers.
00:32:06.220 --> 00:32:17.200
But be careful, look at this. Okay, It is not possible here for x minus 9 to equal 0.
00:32:17.220 --> 00:32:27.150
So we have to exclude x equals 9 from the domain.
00:32:28.000 --> 00:32:38.200
So the domain of this function is this.
00:32:39.100 --> 00:32:44.650
Okay, now we move on to a second set of topics relating to functions.
00:32:47.600 --> 00:32:50.000
Let’s take a look at graphs of functions.
00:32:50.020 --> 00:32:55.450
Remember that the definition of function says that
00:32:55.450 --> 00:33:01.300
one element in the domain is mapped to one and only one element in the range.
00:33:01.700 --> 00:33:04.950
Now how does that apply to the graph of a function?
00:33:04.970 --> 00:33:08.000
Well, let's see. We have eight graphs here,
00:33:08.020 --> 00:33:10.650
and we want to determine if these are graphs of functions.
00:33:11.000 --> 00:33:12.350
So let’s see what we can come up with that
00:33:12.350 --> 00:33:15.900
might help us to do this efficiently.
00:33:16.000 --> 00:33:27.750
Well, let’s see for example any value of x that I select here will map to one value of y.
00:33:27.770 --> 00:33:32.000
Okay, so for example if I pick an x here,
00:33:34.100 --> 00:33:37.200
then there is the ordered pair (x,y) on the graph.
00:33:38.700 --> 00:33:41.900
You can see that on this particular example,
00:33:41.920 --> 00:33:45.250
any x you pick will have one value of y.
00:33:45.270 --> 00:33:50.000
So yes, this is a function. Okay, let’s look at the second one.
00:33:50.800 --> 00:33:55.300
What happens here is that you have only one value of x.
00:33:55.320 --> 00:33:59.100
This is a vertical line and the value of x is the same on this line,
00:33:59.120 --> 00:34:01.600
and so what happens is you might have a point
00:34:01.620 --> 00:34:06.650
here that is going to be (x,0), and then up here somewhere you have exactly
00:34:06.670 --> 00:34:10.450
the same x but maybe the y value is 5.
00:34:11.100 --> 00:34:14.650
So what you can see here is that a vertical line
00:34:14.670 --> 00:34:16.850
does not represent the graph of a function.
00:34:17.300 --> 00:34:20.250
So the answer to b is, is this a function? No.
00:34:20.270 --> 00:34:22.800
But let’s get a little bit more out of this.
00:34:22.820 --> 00:34:28.300
Let’s realize here that a vertical line is never going to represent a function.
00:34:28.320 --> 00:34:32.150
Not only that, we could use a vertical line to help us
00:34:32.170 --> 00:34:34.750
determine whether or not a graph is a function.
00:34:34.770 --> 00:34:37.700
So let’s go back to a and see how this would work.
00:34:37.720 --> 00:34:42.000
If you visualize vertical lines anywhere on the graph
00:34:42.020 --> 00:34:45.250
you can see that any vertical line that you would draw
00:34:45.270 --> 00:34:47.950
would touch this graph only one time.
00:34:47.970 --> 00:34:52.350
What happens is, here, when you draw a vertical line,
00:34:52.370 --> 00:34:57.650
it touches this graph in an infinite number of points, all the points on the line.
00:34:57.670 --> 00:35:01.350
What results here is what is called the vertical line test.
00:35:01.370 --> 00:35:06.350
Okay, we can just call this VLT for vertical line test. Here is how it works.
00:35:07.600 --> 00:35:13.200
If you can draw a vertical line that crosses a graph more than once,
00:35:13.220 --> 00:35:17.850
or touches more than once, then that is not the graph of a function.
00:35:18.150 --> 00:35:20.300
Okay, so let’s try it on part c.
00:35:21.000 --> 00:35:23.700
Notice that no matter where we draw a vertical line,
00:35:23.720 --> 00:35:25.900
it touches the graph only one time.
00:35:25.920 --> 00:35:30.850
So yes, this horizontal line represents the graph of a function.
00:35:31.750 --> 00:35:34.550
Okay, part d. As we draw vertical lines,
00:35:34.570 --> 00:35:37.350
we can see that this vertical line crosses,
00:35:38.000 --> 00:35:41.250
or any vertical line crosses the graph only once.
00:35:42.950 --> 00:35:48.200
Yes, this represents a function. Okay, what about a circle?
00:35:48.900 --> 00:35:53.350
Well anywhere between here and here that we draw a vertical line,
00:35:53.370 --> 00:35:57.300
you can see the vertical line is going to cross the graph twice,
00:35:57.320 --> 00:36:01.600
and that is because for the same value of x you have two different y-values.
00:36:01.620 --> 00:36:03.950
That violates the definition of a function.
00:36:03.970 --> 00:36:09.800
So no, circles are not going to represent graphs of functions.
00:36:09.820 --> 00:36:14.000
Once again, vertical lines. Anywhere that you draw the vertical line,
00:36:14.020 --> 00:36:17.400
you get only one cross. This is the graph of a function.
00:36:18.200 --> 00:36:22.900
Oops, what happens here? Anywhere from here on out,
00:36:23.800 --> 00:36:28.500
when you draw a vertical line, the vertical line crosses the graph twice.
00:36:28.520 --> 00:36:34.100
So you have, right here, you have x’s with two different y-values.
00:36:34.120 --> 00:36:36.300
This does not represent a function.
00:36:37.350 --> 00:36:42.850
The last one: any vertical line crosses only once,
00:36:42.870 --> 00:36:45.950
and so yes, this is the graph of a function.
00:36:45.970 --> 00:36:50.000
Alright, let’s take a look at more graphs of functions.
00:36:52.500 --> 00:36:56.850
Let’s use the graph of this function, f, to answer the following questions.
00:36:56.870 --> 00:37:00.700
Be very careful here about the notation that you use.
00:37:01.500 --> 00:37:06.000
Part a, what is f of 4? Alright, now what does this mean?
00:37:06.020 --> 00:37:09.500
This means that we want x to be 4,
00:37:09.500 --> 00:37:13.950
and we are looking for the y-value on the graph when x is 4.
00:37:13.970 --> 00:37:18.600
In other words, we are looking for the value of the function at 4.
00:37:18.620 --> 00:37:25.850
Okay, when x is 4, we look at this area right here.
00:37:25.870 --> 00:37:29.000
x is 4 anywhere up and down this line.
00:37:29.020 --> 00:37:42.850
On the function we see here that x is 4 when y is 3. So f of 4 is 3.
00:37:43.500 --> 00:37:50.000
Do not put the ordered pair (4,3). Just put 3, the y-value.
00:37:50.620 --> 00:37:55.000
Okay, part b is f of negative 4 positive or negative?
00:37:55.020 --> 00:37:58.950
Well f of negative 4 is a y-value.
00:37:58.970 --> 00:38:02.800
It is the y-value of the function when x is negative 4.
00:38:02.820 --> 00:38:09.000
Okay, let’s see. x is negative 4 anywhere up and down this line,
00:38:09.020 --> 00:38:13.800
and on our specific graph, right there x is negative 4.
00:38:13.820 --> 00:38:20.000
You can see that that value is below the x-axis,
00:38:20.020 --> 00:38:24.100
and so the y-value here, it looks like it is approximately negative 1.
00:38:24.120 --> 00:38:26.000
But we were not asked for the value,
00:38:26.020 --> 00:38:33.300
we were just asked if it is positive or negative. So we are going to put negative.
00:38:33.450 --> 00:38:38.800
Again, approximately negative 1, but that is the correct answer for the specific question.
00:38:38.820 --> 00:38:45.850
Alright, part c: how often does the line x equal 2 intersect the graph?
00:38:45.870 --> 00:38:48.250
Well, what does the line x equal 2 look like?
00:38:48.900 --> 00:38:53.900
Well, x equal 2 is going to be a vertical line.
00:38:54.850 --> 00:38:58.000
Let’s see how many times does it intersect the graph?
00:38:58.020 --> 00:39:02.400
Once. Think about the vertical line test and remember
00:39:02.420 --> 00:39:06.000
that since this a function that is exactly what you would expect:
00:39:06.020 --> 00:39:10.000
a vertical line would only intersect this graph once, and that is what happens.
00:39:12.800 --> 00:39:18.500
Okay, the domain. This is the set of all x’s for which the function is defined.
00:39:18.520 --> 00:39:21.850
The smallest value of x is negative 5.
00:39:21.870 --> 00:39:30.850
As we read from left to right, we see the largest value of x is 9. Both endpoints are included.
00:39:31.350 --> 00:39:38.900
In interval notation, we are going to write the domain this way.
00:39:39.200 --> 00:39:43.650
Okay the range is the set of all y values for which the function is defined.
00:39:43.670 --> 00:39:48.800
So as we read from bottom up to top, smaller to larger values of y,
00:39:48.820 --> 00:39:52.950
we can see that that the smallest value of y is negative 2
00:39:52.970 --> 00:39:56.000
here and negative 2 here. So that is the smallest value.
00:39:56.020 --> 00:40:10.400
The largest value of y is 4. So the range goes from a value of negative 2 to 4.
00:40:10.420 --> 00:40:14.900
Again both endpoints are included and so I put brackets.
00:40:15.550 --> 00:40:22.950
Okay, part e: for what values of x does f of x equal 4?
00:40:23.300 --> 00:40:28.850
Okay now, this number right here, this is a y-value.
00:40:28.870 --> 00:40:36.900
So what this says is when y is 4, what is x?
00:40:36.920 --> 00:40:44.150
Okay, well y is 4 anywhere along this line that goes right through here.
00:40:45.000 --> 00:40:53.150
You can see that the only x-value on the function where y is equal to 4 is 2.
00:40:53.170 --> 00:40:55.500
So we are going to write down the number 2.
00:40:56.900 --> 00:41:01.000
It would not be correct to write the ordered pair (2,4).
00:41:01.020 --> 00:41:07.000
The question says what is x when y is 4. Answer: 2.
00:41:08.150 --> 00:41:13.350
Okay, part f is kind of a similar question.
00:41:13.370 --> 00:41:20.950
It says when f of x is 3, what is x? Okay, that means when y is 3.
00:41:21.000 --> 00:41:25.500
Okay, y is 3 right along this line,
00:41:25.520 --> 00:41:27.300
and you can see that on our function
00:41:27.320 --> 00:41:34.850
there are two places where the graph has a y value of 3.
00:41:35.600 --> 00:41:39.200
Here when x is 0, and also when x is 4.
00:41:39.900 --> 00:41:42.400
Okay, so I am going to list these two answers.
00:41:42.420 --> 00:41:44.650
Of course you have to read the directions on a problem,
00:41:44.670 --> 00:41:49.150
and be careful that you answer the question exactly as it is asked.
00:41:49.170 --> 00:41:50.950
But in this case I am just going to
00:41:50.970 --> 00:41:54.400
list the two values 0 and 4, no parenthesis.
00:41:54.420 --> 00:41:59.000
This is not an ordered pair, these are values of x, this 0 and this 4.
00:42:00.000 --> 00:42:03.950
Okay, last question: what are the x-intercepts?
00:42:03.970 --> 00:42:09.100
Now x-intercepts are the actual values of x
00:42:09.120 --> 00:42:12.000
where the graph crosses the x-axis.
00:42:12.000 --> 00:42:15.150
Sometimes intercepts are listed as ordered pairs,
00:42:15.150 --> 00:42:21.000
but in this when you are just asked to list the x-intercepts,you list just the x-values.
00:42:22.000 --> 00:42:22.450
So we would have an intercept here
00:42:22.470 --> 00:42:27.750
at negative 3, one at 6, and another one at 9.
00:42:27.770 --> 00:42:33.700
So I would put negative 3, 6, and 9.
00:42:34.200 --> 00:42:36.650
Okay, if I were to list these as ordered pairs,
00:42:36.670 --> 00:42:43.300
I would put (-3,0), (6,0), and then (9,0).
00:42:43.320 --> 00:42:46.550
The y-intercept. Of course this is a function
00:42:46.570 --> 00:42:49.100
so there is only going to be one y-intercept.
00:42:49.120 --> 00:42:52.600
Here it is: it occurs at the point (0,3),
00:42:52.620 --> 00:42:58.100
and so I am going to list the y-intercept, and the y-value is 3.
00:42:59.100 --> 00:43:03.650
Alright, let’s take a look at a graph of another function.
00:43:06.100 --> 00:43:08.400
Let’s answer these questions about the function
00:43:08.420 --> 00:43:11.650
f of x equals 3 x squared minus x minus 2.
00:43:12.300 --> 00:43:14.250
This is the function we are going to work with.
00:43:14.270 --> 00:43:18.100
Part a asks is the point (2,8) on the graph of f?
00:43:18.120 --> 00:43:25.000
This means is f of 2 equal to 8? Well, let’s test.
00:43:25.020 --> 00:43:41.800
f of 2 would equal 3 and then 2 squared minus 2 minus 2.
00:43:41.820 --> 00:43:45.400
Let’s check that. I substituted 2 into here and here.
00:43:45.420 --> 00:43:50.000
So this is f of 2 and my question is: is it equal to 8?
00:43:50.020 --> 00:43:54.900
2 squared is 4. 4 times 3 is 12.
00:43:54.920 --> 00:43:59.350
So I get 12 minus 2 minus 2, which is minus 4,
00:43:59.370 --> 00:44:06.900
and I get 8. So what that means is that f of 2 is equal to 8,
00:44:06.920 --> 00:44:11.000
which means that the point (2,8) would be on the graph of this function.
00:44:11.020 --> 00:44:14.850
So the answer is yes.
00:44:15.550 --> 00:44:20.850
Part b: if f of x is negative 2, what is x?
00:44:21.350 --> 00:44:26.150
Okay, this means that y is equal to negative 2.
00:44:26.170 --> 00:44:30.850
Now y is f of x, y on the graph is f of x here.
00:44:30.870 --> 00:44:35.100
So I am going to substitute in place of f of x negative 2.
00:44:35.120 --> 00:44:40.950
So I am going to have negative 2 equals, and then this expression,
00:44:40.970 --> 00:44:46.700
3 x squared minus x minus 2.
00:44:47.250 --> 00:44:53.200
Okay, wen I add 2 to both sides, I get 0 here and 0 here.
00:44:53.220 --> 00:44:56.750
I am going to reorder this, make it easier to look at.
00:44:56.770 --> 00:45:02.600
I am going to write 3 x squared minus x equals 0.
00:45:02.620 --> 00:45:09.000
I want to solve for x. I am going to factor an x out. That leaves me
00:45:09.020 --> 00:45:13.950
3x minus 1 equals 0 for the other factor.
00:45:13.970 --> 00:45:16.100
I am going to set each factor equal to 0.
00:45:16.120 --> 00:45:19.150
Don’t lose this factor, that is a common error.
00:45:19.170 --> 00:45:25.450
I am going to get either x equal 0 or 3x minus 1 equal 0,
00:45:25.470 --> 00:45:37.000
which means x equals 1/3. So what that means is that when f of x is negative 2,
00:45:37.020 --> 00:45:42.950
there are two values of x: 0 and 1/3.
00:45:43.200 --> 00:45:47.350
This is not an ordered pair, this is a listing of two x-values.
00:45:47.370 --> 00:45:50.250
Then the c part is to state the domain.
00:45:50.900 --> 00:45:56.000
Well this function has no restriction on the domain.
00:45:56.020 --> 00:45:58.750
There is no reason why you could not use any real number
00:45:58.770 --> 00:46:02.700
for x substituted into here and get a value for f of x.
00:46:02.720 --> 00:46:05.000
So the domain is all real numbers.
00:46:05.020 --> 00:46:09.400
In interval notation, it would be written this way.
00:46:09.420 --> 00:46:18.000
Or depending on the directions you could put a capital R symbol for real numbers.
00:46:19.350 --> 00:46:23.900
Alright, you need to study these examples carefully,
00:46:23.900 --> 00:46:26.650
and be sure that you have mastered these skills.
00:46:26.670 --> 00:46:30.000
These are very important skills about graphs of functions.
00:46:32.800 --> 00:46:37.650
Part d: list the x-intercepts. The x-intercepts are going to be
00:46:37.670 --> 00:46:43.650
the x-values where y, or f of x, equals zero.
00:46:44.300 --> 00:46:48.000
So let’s let f of x equal 0, and we are going to write
00:46:48.020 --> 00:46:55.200
3x squared minus x minus 2, and what we want to know is
00:46:55.220 --> 00:46:59.000
what values of x makes this equal to 0.
00:46:59.020 --> 00:47:00.950
Now, this is a quadratic equation
00:47:00.970 --> 00:47:04.350
so the first thing we are going to do is attempt to factor it.
00:47:04.370 --> 00:47:09.850
Let’s see if it factors easily. It might.
00:47:10.150 --> 00:47:12.950
We can see that we need 3x then x,
00:47:12.970 --> 00:47:15.750
and then the factors of 2 will be 2 and 1,
00:47:16.700 --> 00:47:22.950
and so if we put a 2 here and a 1 here,
00:47:23.000 --> 00:47:25.900
and then we want the 3x to be negative
00:47:25.920 --> 00:47:29.900
so I put a negative here and positive here.
00:47:29.920 --> 00:47:33.600
Let’s check it. That is correct.
00:47:33.620 --> 00:47:35.300
We set each factor equal to 0.
00:47:35.320 --> 00:47:46.700
So we get 3x plus 2 equals 0. x is negative 2 divided by 3.
00:47:46.720 --> 00:47:50.850
Then here we get x minus 1 is 0.
00:47:50.870 --> 00:47:57.950
x equals 1. So we have two values of x were f of x equals 0,
00:47:57.970 --> 00:48:00.550
and these are our two x-intercepts.
00:48:00.570 --> 00:48:06.900
I am going to list them. These are not ordered pairs.
00:48:07.000 --> 00:48:10.000
If they were ordered pairs, they would be listed as
00:48:10.020 --> 00:48:15.100
(-2/3,0), one pair, and (1,0) as the other pair.
00:48:15.900 --> 00:48:23.100
Okay, now the y-intercept is going to be the value of y where x is zero.
00:48:23.120 --> 00:48:27.150
So that means I am going to find f of 0.
00:48:28.950 --> 00:48:32.150
This is going to be pretty simple because f of 0 is going to be
00:48:32.170 --> 00:48:36.900
0 for this term, 0 for this term, and what I have left is negative 2.
00:48:37.600 --> 00:48:44.000
That is of course the one and only y-intercept, negative 2.
00:48:44.020 --> 00:48:51.000
Alright, part e: if x is negative 1, what is f of x?
00:48:51.020 --> 00:48:58.200
Well, this is just another way of asking what is f when x is negative 1.
00:49:00.750 --> 00:49:07.950
Okay, and so I am going to substitute negative 1 into here.
00:49:08.550 --> 00:49:12.750
Negative 1 in for the other x, and then
00:49:12.770 --> 00:49:15.950
I am going to simplify the expression on the right.
00:49:15.970 --> 00:49:24.200
This is positive 1 times 3, so this would be 3 plus one minus 2,
00:49:25.000 --> 00:49:35.100
which is 4. So f of x is equal to 4 when x is negative 1.
00:49:35.120 --> 00:49:38.900
This is, let’s see, I think maybe that should be 2.
00:49:40.850 --> 00:49:48.300
It would be 4, I better finish this, 4 minus 2 and so this should be 2.
00:49:48.320 --> 00:49:51.850
Always be careful and check your work.
00:49:51.870 --> 00:49:55.200
Alright, you need to carefully practice.
00:49:55.220 --> 00:49:58.350
Study these examples until you have attained
00:49:58.370 --> 00:50:04.900
a very high skill level with this information.
00:50:05.100 --> 00:50:10.000
Example four: use the graph of the function to answer these questions.
00:50:10.300 --> 00:50:12.3500
So here we have the graph of a function,
00:50:13.000 --> 00:50:21.700
and part a says is f, the function, increasing on the interval from 0 to 5?
00:50:21.720 --> 00:50:24.300
Alright, now let’s take a look at this question carefully.
00:50:24.320 --> 00:50:28.000
First of all, this is not an ordered pair (0,5),
00:50:28.020 --> 00:50:31.950
this is an interval as x goes from 0 to 5.
00:50:31.970 --> 00:50:34.200
So let’s look at where this is on the graph.
00:50:34.220 --> 00:50:40.500
Okay, well x equals 0 starts here and x equals 5,
00:50:40.520 --> 00:50:42.900
three, four, five, goes to about here.
00:50:43.500 --> 00:50:49.000
So what we are doing here is we are looking at the interval from 0 to 5.
00:50:49.020 --> 00:50:52.950
When you talk about increasing or decreasing you always,
00:50:52.970 --> 00:50:56.850
always, read from left to right.
00:50:57.450 --> 00:50:59.950
Do not pay any attention to what the arrows
00:50:59.970 --> 00:51:05.600
on the ends of the graph may cause your eye to do. You read from left to right.
00:51:06.020 --> 00:51:08.900
Alright, so over the interval from 0 to 5
00:51:08.920 --> 00:51:10.650
let’s look at what the graph is doing.
00:51:11.100 --> 00:51:18.200
It is decreasing, increasing, decreasing, and then increasing to about right here.
00:51:18.220 --> 00:51:21.850
Is f increasing on the interval from 0 to 5?
00:51:21.870 --> 00:51:28.000
Clearly the answer is no. It is doing some increasing and some decreasing.
000:51:29.150 --> 00:51:33.200
Now, the b part gets more specific.
00:51:33.220 --> 00:51:38.550
It says list the intervals, intervals: values of x,
00:51:38.570 --> 00:51:42.000
on which f, the function, is increasing.
00:51:42.550 --> 00:51:46.600
Then the c part is going to be the same thing but decreasing.
00:51:46.620 --> 00:51:50.000
So what we are going to do is we are going to read from left to right,
00:51:50.020 --> 00:51:53.650
and we are going to identify all of the intervals
00:51:53.670 --> 00:51:56.000
where the function is increasing and decreasing.
00:51:56.020 --> 00:51:58.000
Then we are going to back and write the answers down.
00:51:58.020 --> 00:52:01.700
Alright, so as I read from left to right,
00:52:01.720 --> 00:52:07.200
I can see that the graph is decreasing as x goes from
00:52:07.220 --> 00:52:13.950
actually negative infinity to the x value negative 5.
00:52:13.970 --> 00:52:22.000
So right here, the function is decreasing,
00:52:22.950 --> 00:52:28.000
and then the function turns, and begins to increase.
00:52:28.020 --> 00:52:33.650
It increases until x gets to negative 2.
00:52:34.800 --> 00:52:45.150
Then as x goes from negative 2 to 1, the function is decreasing.
00:52:46.150 --> 00:52:54.950
Then as x goes from 1 to 2, the function is increasing.
00:52:55.600 --> 00:53:00.900
Then from 2 to 3, the function is decreasing.
00:53:02.250 --> 00:53:07.950
Then from 3 to infinity the function is increasing.
00:53:08.000 --> 00:53:10.950
Alright, now let’s see if we can get all these intervals written down.
00:53:10.970 --> 00:53:13.850
Remember that these are values of x.
00:53:13.870 --> 00:53:18.700
You do not write ordered pairs, you write intervals on x.
00:53:18.720 --> 00:53:24.900
Alright so from negative infinity to negative 5, the function is decreasing.
00:53:29.100 --> 00:53:34.000
We are going to use open intervals for all of these answers.
00:53:34.020 --> 00:53:45.150
Then from negative 5 to negative 2, the function is increasing.
00:53:46.800 --> 00:53:56.900
Okay, then from negative 2 to 1, the function is decreasing.
00:53:58.000 --> 00:54:06.900
Negative 2 to 1. Then from 1 to 2, the function is increasing.
00:54:07.000 --> 00:54:13.900
From 2 to 3, the function is decreasing.
00:54:14.000 --> 00:54:20.200
From 3 to infinity, the function is increasing.
00:54:21.000 --> 00:54:24.250
Now let’s take a look at exactly what we have here.
00:54:24.650 --> 00:54:29.600
We should have all of the elements of the domain except
00:54:29.620 --> 00:54:33.300
for the actual endpoints where the function changes from
00:54:33.320 --> 00:54:37.900
increasing to decreasing. So we have negative infinity to
00:54:37.920 --> 00:54:41.900
negative 5, negative 5 to negative 2, negative 2 to 1,
00:54:41.920 --> 00:54:45.650
1 to 2, 2 to 3, and 3 to infinity.
00:54:45.670 --> 00:54:48.150
So we have covered all of the intervals.
00:54:48.170 --> 00:54:51.000
Let’s take a look at a few more questions.
00:54:53.100 --> 00:54:58.950
Part d: determine whether this function is even, odd, or neither.
00:54:59.150 --> 00:55:01.100
The simplest way to determine this is to
00:55:01.120 --> 00:55:03.000
remember what you know about symmetry.
00:55:03.900 --> 00:55:08.750
An even function is symmetric with respect to the y-axis.
00:55:08.770 --> 00:55:12.200
An odd function is symmetric with respect to the origin.
00:55:12.750 --> 00:55:17.900
So let’s take a look. Look at the purple graph of the function.
00:55:18.550 --> 00:55:23.300
Do you see and y-axis symmetry or any origin symmetry?
00:55:23.320 --> 00:55:26.850
The answer is no, and so we can say that
00:55:26.870 --> 00:55:35.900
this function is neither even nor odd.
00:55:37.000 --> 00:55:44.900
Part e: list the values of x at which f has a local maximum.
00:55:45.800 --> 00:55:49.150
Well let’s see, where would we find a local maximum?
00:55:49.850 --> 00:55:54.700
A local maximum would occur where a function changes
00:55:54.720 --> 00:55:57.850
from increasing to decreasing.
00:55:57.870 --> 00:56:00.200
So right there, that is a local maximum.
00:56:00.220 --> 00:56:03.000
Then you see another one occurs here where
00:56:03.020 --> 00:56:07.150
the function changes from increasing to decreasing.
00:56:07.170 --> 00:56:12.850
The values of x where this occurs would be listed as
00:56:12.870 --> 00:56:20.600
negative 2, and 2. This is not a point,
00:56:20.620 --> 00:56:23.600
this is not an interval, these are values of x
00:56:23.620 --> 00:56:28.350
where we have a local max. What are these local maxima?
00:56:28.370 --> 00:56:35.750
This local max is 4, and this local max is 3.
00:56:36.600 --> 00:56:39.000
Notice that I do not put ordered pairs here,
00:56:39.020 --> 00:56:44.150
I simply put the y-values because the actual max is the y-value,
00:56:44.170 --> 00:56:48.400
or the value of the function at these particular values of x.
00:56:49.000 --> 00:56:52.700
Okay, same question but about a min.
00:56:52.750 --> 00:56:57.900
So list the values of x at which the function has a local min.
00:56:58.500 --> 00:57:02.650
Alright, a local min is going to occur where a function changes
00:57:02.670 --> 00:57:07.800
from decreasing to increasing. So one here.
00:57:08.200 --> 00:57:10.450
Decreasing to increasing, one here.
00:57:10.470 --> 00:57:14.250
Decreasing to increasing, so we are going to have three values
00:57:14.270 --> 00:57:17.900
of x where we have a local min occur.
00:57:18.400 --> 00:57:25.000
Those values would be negative 5, 1, and 3.
00:57:26.350 --> 00:57:32.700
Negative 5, 1, and 3, those are the values of x were a local min occurs.
00:57:32.720 --> 00:57:36.000
Now the local min itself here is negative 3.
00:57:36.950 --> 00:57:42.950
The local min here is 0. The local min here is also 0.
00:57:42.970 --> 00:57:48.650
So we are going to list the negative 3, and then the 0.
00:57:48.670 --> 00:57:52.150
Even though this occurs twice, we will list it once.
00:57:53.000 --> 00:57:58.500
Practice working with these questions and these examples
00:57:58.520 --> 00:58:00.100
until you have mastered the skill.