Calculus I - Lecture 02

as we continue looking at functions now we'll look at how you create new functions from old and we'll start with operations on functions now let's suppose we have a couple of functions so let's let F and G be functions the first way you can combine F and G to create new functions of course is to use the arithmetic operations which is the same operations you use on real numbers arithmetic operations and they result in the following new functions F plus G applied to X would give you f of X plus G of X another new function f minus G of X would give you f of X minus G of X another new function again using these arithmetic operations would be F times G of X and that's going to be f of X times G of X and then finally F over G of X the quotient and this one will have a proviso of course f of X over G of X and this will be provided the bottom G of X is not 0 because we can't divide by 0 so these are the arithmetic operations these produce new functions now there's actually something hidden here that needs to be taken into account in the beginning F and G have their own domains but when they are combined in this way the domain of the combination has to be the intersection of those two domains so it is the common domain to the two different functions that you started with something to keep in mind and we'll talk about that more in just a minute the other operation which stands outside of the arithmetic operation is called composition of functions and it's different because you can't do this with real numbers this is something you can only do with functions and here is the idea and then I'll write the definition on the next page the idea is as follows with functions if you start with an X say and you apply a function called G to it and you get a G of X nothing new there that's just a function but with a function you can apply a second function in sequence that second function perhaps would be called F and you get F of G of X in the end and then this entire function will be called F composed with G and that's an open hole and that will be the new function here that is composed of the old functions so let's go ahead and look at the proper definition of that the proper definition is that F composed with G of X and again that's the open hole as opposed to the closed hole which is used for multiplication the best way to write this though is to think of it as F of G of X that function where the domain of the composition is equal to the set of all X values in the domain of G because when you apply this function remember the X's are starting at the far left their far starting with the G set of all X in the domain of G such that G of X the result of applying G to X ends up being in the domain of F and if you look at the picture we had before you see that X goes to G of X and then F has to act upon this G of X so whatever X's are allowed here must be ones that produce G of X's that F can then act upon so that gives us our definition of the composition of two functions and here is the note I mentioned earlier but I will now write it new functions created in whatever form here always are going to inherit some sort of domain restrictions they will always inherit domain restrictions and let me give you an example of that with one of the functions that was not a composition of function say the product of two functions if say f of X is equal to the square root of X and G of X is likewise equal to the square root of x so they both are the same function let us find F times G of X so this is the product of these two functions well the solution to this is to multiply them of course but since we want to concentrate on the domains let's first look at the domains of the two functions the domain of F is equal to what well the square roots only defined for 0 and positive numbers so this would be 0 to infinity and that of course is the same as the domain of G since they are the same function then look at what happens then if I take F times G of X which is the product here of these two functions f of X times G of X here that's the square root of x times the square root of x which is X now if you think of X alone as a function it has no restrictions on its domain however here it's going to inherit that the restrictions that it brings with it from the composition the product in this case of square root of x and square root of x so if I write down the domain of F times G I'm going to get 0 to infinity including 0 this is the same inherited domain in fact we get the whole domain that we inherited this time sometimes we only get a part of it but in this case this is the domain that this function is restricted to even though if you saw a function equal to X with no information you think its domain with all the real numbers here however it's restricted because of where it came from now let's look at another example where we can try composition f of X is x squared plus 3 say and G of X is again say the square root of x what we want to do here is find the composition F composed with G and the composition in the other way G composed with F now let's look at these two domains they will come into play so let's go ahead and mark them while we're here the domain of F is all real numbers you can square and add 3 to any real number the domain of G as we already know is limited to 0 to infinity including 0 ok so if we want to compute these two compositions and these are open holes so these are compositions let's start with the first one F composed with G of X well by definition that's F of G of X G of X we can just copy down this is f of the square root of x and now comes the stage where you have to understand what composition is in order to take the next step f acts upon this object the way it acts upon all objects F squares an object and adds 3 to it so here we have the square root of x so f is going to square the square root of x leaving us with x plus 3 so you see this was squared and 3 was added to it this is squared and 3 was added to it so that's X plus 3 so that is F composed with G and the domain of F composed with G let's see it's restricted by the domain of G because that's where the X is start from and then what kind of restriction does F add to it will nothing so the domain is simply what it was for G now if we reverse the composition let's see what happens this would be G of f of X and again f of X is right here so we can copy it down here's G of x squared plus 3 here again is the step that shows whether you understand composition what does G do to something well here's G G takes the square root of something so here this is going to give us the square root of in this case x squared plus 3 so here is our function G composed with F first lesson here is of course that these are not the same and in general you shouldn't expect them to be the same composition in two different directions ought to produce most of the time something different what is the domain just to finish this off of this second composition well it certainly is going to include it's going to begin with the domain of F the domain of F happens to be everything so there's no restrictions then g1 applied to it produces this so the domain here is going to be not restricted by F at all but it will be restricted by G in what way this expression under here can only be 0 or positive so the domain of this is going to be where X has x squared plus 3 greater than or equal to 0 there's a verbal description of that domain now we'd like to get a better description of that so we can actually find that out and we can go ahead and check that but wait x squared plus 3 is always positive 3 is positive x squared is either 0 or positive so this is always positive so there's no check needed here so in fact any X will work and this is all real numbers so it looked like we might have work to do there but in fact we don't in this case and let's just stop there that will be a couple of good examples of how functions are combined and we'll go on and look at something else let's ask ourselves how these operations on functions affect the function graphs so let's go ahead and look at that for our example here just so we have a function to look at let's just look at f of x equals x squared we'll keep that as our simple example here the first operation that you can perform on functions is an operation that results in the translation of the graph and we'll take C greater than 0 here and you'll see where the sea comes in in just a minute so for translation if you want the function graph to move left then you compute f of X plus C that will cause the function to move to the left now the original function here I guess I should draw it here so we remind ourselves there's the original parabola and it is shifted left by C we're assuming C is positive here if you compute f of X plus C so if you replace X everywhere by X plus C this is what will happen to the picture if you want to move right you do f of X minus C and in the same way you will end up with the parabola move to the right if you want to go up you just take the original y-value f of X and add C to it and what that will do is it will push the original parabola up by C and if you want to go down as you might expect f of X minus C and that will take your original parabola and shift it downward to minus C so those are the operations you can perform to a function to affect its graph in these four ways there is another way that you can affect the graph these four ways are called translations you're not rotating or turning the graph you just shifting it right or left or up or down you can also do reflection if you take and calculate f of minus x what that does is take the original picture and shift it across the y axis now there's no change here but that's only because it is the x squared function usually there will be a difference when of course you reflect it across the Y axis the other thing you can do for reflection if you take the negative of the original function this one's easier to see the function just flips over so this is an up/down reflection and that flick flips you across the x-axis what else can we do well we could do stretches or compressions and these are accomplished by multiplication if you multiply the original function f of X by C and C is a number bigger than 1 then for example if C is 2 it's going to double all the Y values so that's certainly going to be a stretch and it's going to stretch it since we're doubling Y values this would be a vertical stretch and so in our picture for the parabola we're going to get a long skinny parabola that's been stretched vertically if you multiply C times f of X again but this time you take your positive C between 0 & 1 you get a vertical compression so what happens is the parabola now becomes low and wide and we're skipping one here because that doesn't change the function at all of course if you multiply the x value by C first and then apply the function to it where C is greater than 1 this time you have the effect that is horizontal and this will be horizontal compression and in this case horizontal compression looks exactly like a vertical stretch for a parabola so it's actually going to look just the same in this particular case and if you apply F to C X where C is between 0 & 1 you get a horizontal stretch so it stretches in the right-to-left direction and in that case it's going to look a lot like vertical compression due to the nature of this function so this is the kind of stretch that you would get so those are operations these stretches the compressions the translations and the reflections that cause certain kinds of things to happen to the graph let's do an example of the sort of thing that you ought to be able to do sketch the following function y equals 4 minus the absolute value of X minus 2 the solution here is to realize that this is a combination of operations that affect the graph of the core function which is the absolute value function here so we can do this pictorially and see what's happening if we start with the absolute value function that's that 45-degree line y equals absolute value of x and then we work our way forward first of all change it to the absolute value of x minus 2 what does that do that is going to be a translation as we just saw that translates the curve to the right by so this is y equals the absolute value of X minus 2 then if we multiply that by a minus 1 what happens is that we reflect the curve through the x-axis this is still to here but now it's facing downward so this is y equals minus the absolute value of X minus 2 and finally we need to add this 4 and that will shift the curve up so we'll still have this previous shape this downward pointing arrow and we will need to move it up centered here at 2 and it will go up to a height of 4 so will look something like this when we're done and this is the curve y equals 4 minus the absolute value of X minus 2 that we were trying to graph and so this illustrates what you should be able to do if you start with some sort of core function in this case the absolute value function could be x squared could be any of a number of other functions and then apply certain operations to it and this is how the graphs are affected now one more item regarding graphs functions with symmetric graphs there are two sorts of functions that are very common and we will see them as we move along some of them will be trigonometric functions some of them will be powers of X and here are the definitions that will be appropriate f is called an even function if when you take F and you evaluate it at minus X so this is the test if you evaluate X at F rather at minus X and you get back f of X as though there were no change you have what's called an even function what is interesting about that well what's interesting about it is that this sort of a function is symmetric with respect to the y-axis so if this is your axis system this curve is symmetric with respect to the y axis is it is the same on either side is reflected as in a mirror the other definition that is useful to us is f is odd and what would that be well again you take exactly the same test case you take F of minus X you put minus X in wherever X is if in the end you end up with minus the original function instead of here with even where you just got the original function if you have the negative this produces a function whose graph is symmetric with respect to the origin now that means that if you go across the origin you have symmetry let me show you what an example of that might be if we take a function like f of x equals x cubed and this is where the name came from odd powers were the first examples of odd functions although they're not the only ones if you look at the graph of this from your own experience you have a graph that looks like this and notice that if I have a point over here that there is a matching point here that is on a line through the origin well that is exactly the characteristic that defines a function which is symmetric with respect to the origin and so both of these sorts of functions even and odd are going to occur again and again and it's good to have these definitions straight time for a couple of exercises our first exercise will begin with the function f of x equals x squared and the second function G of x equals say the square root of 1 minus X what we want to do here is find the composition F composed with G and the composition G composed with F and their domains so go ahead and calculate those compositions work out their domains and we'll be back let's see how these compositions worked out first I'm going to look at the domains of the two functions just so I have that information when I want to compute the domains of the composition it'd be nice to know that so let us note that the domain of the original function f which is x squared is all of the real numbers because you can square any real number and the domain of G well that might take a little more work the domain of G is going to be where 1 minus X is greater than or equal to 0 because 1 minus X is under the square root and the square roots only defined for numbers that are 0 or positive where 1 minus X is greater than or equal to 0 well let's see what that would lead to that means 1 must be greater than or equal to X that tells me exactly what X should be so that means the domain of G ought to be the interval from minus infinity to 1 including one that would be exactly X less than or equal to 1 so I have my two domains there and let me mark them off so I don't lose them and let's go ahead now and look at the compositions so F composed with G is f of G of X and again I can copy G of X straight down from here so this is F of the square root of 1 minus X and here is the key quality where I see if I understand composition what is f do to something F squares that's something so if F squares the square root this is going to leave me with 1 minus x so this is 1 minus X what would be the domain of this function the domain of F composed with G well let's see the original function here is G G has a restricted domain which was this interval and then F acts upon it and half squares it f has no restrictions so the domain of this function although one minus x has no restrictions if it stands by itself since it is the product of this composition it inherits the Direction's that lie in there and so this domain is going to be minus infinity to one as an inherited restriction so this is inherited likewise if we compute G composed with F that's G of f of X that's G of x squared remembering that F is x squared here what does G do to something G takes the square root of 1 minus that something so this is the square root of 1 minus x squared and that is the Equality that tells whether you understand composition so here is the function I end up with the square root of 1 minus x squared what is its domain well there's no restriction inherited from the F part because f was defined here for all real numbers so the only restriction is going to be what G imposes and so this must make sense we need 1 minus x squared to be greater than or equal to 0 well now we just work out the algebra to see what that implies this means that 1 is greater than or equal to x squared and note that if it is x squared x squared is greater than or equal to 0 if we take the square root throughout we get 1 greater than or equal to the square root of x squared greater than or equal to 0 and you may remember and you should that the square root of x squared is the absolute value of x so this is 1 greater than or equal to the absolute value of X greater than or equal to 0 if you look at the number line here's 0 and here's one this says that the absolute value of x is between 0 & 1 so it's over here what does that say about X that says that X itself could go as far down as minus 1 and its absolute value would still lie in here so that tells me that the interval that I'm looking for is going to be minus 1 to 1 as the domain of this function so I hope after your work that that's what you came up with and we will now look at another example in this example it'll involve a sketch so we want to sketch the function y equals x squared plus 2x by completing the square a nice algebraic operation you should remember it's good to practice completing the square and transforming as we did in an example earlier transforming the graph of the core function here which is x squared so give that a try and we'll be back in this problem the first thing we need to do is algebra we need to complete the square so let's write this down y equals x squared plus 2x the complete the square is to come up with the constant that will go there to make this into the perfect square of a binomial so what we're supposed to do is look at the coefficient of x take one half of it which is 1 plus 1 in this case square it and add it now of course you can't change the problem so we must also subtract that away this now by design is equal to X plus 1 quantity squared and we have now this minus 1 so Y has been rewritten so that we see that it is a translated version of y equals x squared and that's what we want it to look at so here's completing the square if we now look at the graph we can see how to get there from the core function y equals x squared so here's the original y equals x squared it is the common parabola and let's walk our way forward first of all let's add 1 to X which results remember in a shift to the left of the parabola 1 so this is y equals x plus 1 squared and one final thing we need to subtract one that will shift this parabola downward so we'll end up with here's the 1 and we're going to shift by minus 1 so that's going to give us a parabola that looks something like this which has its vertex here at 1-1 and this is our function X plus 1 squared minus 1 that's what we needed to do here I hope that that's what you ended up with we examine now some families of functions now family of functions is a set of functions that have the same format where certain parameters change the first family we'll look at is the power function family y equals x to the P power we'll start with one of the simplest measures members of that family y equals x and variations on it actually the more general form would be y equals MX plus B which of course everybody knows is the equation of a line there are two parameters here M and B and as you vary them different things happen to the pictures if M varies so the slope varies but B remains fixed the picture that you get is something that looks like this suppose B is a positive number so I can mark it up here this would be B the slope will vary but B is fixed what does B is the y-intercept of the curve so that means you will have graphs of various sorts looking like this this is the family all the lines that pass through this vertical height of B where the slope can be anything because the slope is allowed to vary so all of these lines pass through the point on the y axis B which is the point with coordinates 0 B that's what happens if M Barrie's here if we allow B to vary so we keep em fixed so the slope is always the same but B varies well B just determines where vertically the line is going to strike the y-axis M being fixed means it's always going to strike at the same slope so the picture that we get is something like this this is all these lines have the same slope which is M and there will be infinitely many of them going up and down all of the ones that have slope M will be indicated by the family if we leave em fixed and let be very so these are two examples of families of functions and sometimes it's helpful to know all the properties of a family so that you can examine the family as a whole and draw certain conclusions let's continue to look at this power family and let us look at the case well we're going to look at the case y equals x to the n now this one we'll choose n greater than or equal to one and an integer and n an integer will be the parameter here first we'll examine X to the N where n is even in that case the graphs of this family look like this first of all notice that if n is even if X is zero you get zero no matter what the N is also if X is 1 or minus 1 you will always get a value of 1 that is give us three anchors if you like for this picture there's one there's minus one all of the curves have to pass through all of those now you know the familiar one effects it if we have x squared here at the N equals two we have the familiar parabola that would pass through here like so but as the ends get larger and larger what will happen is that you get parabolic like looking curves but they will vary some will be higher some can be lower but they will all pass through those three points and that will be the defining characteristic of this family now if we look at y equals x to the N where n is odd in this case the curves look considerably different the first odd case is y equals x and you know what that looks like that is simply the 45-degree line here running through the origin now what happens if you have y equal X cubed or y equals x to the fifth and so on well you get curves that look like this and there are some anchor points here too let's go ahead and mark those again it's the same three zero one here no matter what power you take one two it's always going to be one minus one here when you take minus one to an odd power it becomes negative here when you took minus one to a power it would become positive because it was always an even power so all of the curves are going to pass through these three points so you'll get variations on this of all sorts depending on the power involved but that's a description of that family in a graphical way let's continue to look at this value family if n is an integer and not greater than or equal to one can we say something about that well sure if n is a greater than or equal to one and as an integer but instead of looking at the y equals x to the N curves we saw before let's look at y equals x to the minus n curves so that's like looking at the negative integer powers so this would be 1 over X to the n and first of all again let's look at the N equal even number case if you graph these here are the pictures you get again the same thing happens this and fortunately in this case you don't have this defined at 0 but it is defined it 1 and at minus 1 and because n is even that's always going to give the same value which is a height of 1 so those two points will always be passed through notice that since this is not defined at zero for any power this will always be undefined on the y-axis so the kind of pictures you get our pictures that come in two pieces and they might look something like that or maybe like this and there'll be a lot of them an infinite number in fact of curves that look roughly like this they will all pass through these two points and all have the y axis as an asymptote if you look at y equals x to the minus n which of course is 1 over X to the N for an odd what happens is that again it's not defined at 0 and 1 1 goes up to a height of 1 minus 1 because it's an odd power is going to go down to a depth of minus 1 and the first example of course is the 1 over X curve which everyone knows something like this and then other curves will have similar shapes but maybe come in a little tighter and so on as the power of n goes higher and higher so here are again graphical images of these families and that's a good place to stop our next set of families are the polynomial function and rational function families let's first look at the polynomial functions and I hope these are familiar to front to you from your past experience one of the characteristics of these functions in their graphs is that they have no asymptotes either horizontal or vertical and they have this sort of general form a function is a polynomial function if it looks like this say C naught plus C 1 X plus up to C M X to the M and of course is a positive integer or zero and C n in any case is nonzero so that this is really the highest power what kind of examples do we have here well you should know these from your experience let's look at a few of them n equal 1 if N equals 1 then we have something like this form and that's the line and we've seen lines before and so we have lines that may come at various slopes and passing through various of values along the y axis if N equals 2 we have quadratics or parabolas if you like to call them that here is the standard parabola but that parabola as you know can be shifted up or down it can also be flipped over and so there are various versions of this can be stretched or compressed and equals three these get a little harder to draw as time goes on here if you look at n equal three we're looking at cubics here is a familiar standard cubic that cubic can also come up very high and go down very low and like the others it can be shifted it can also go in the other direction so these are variations you can see there are two humps here which is sort of a defining characteristic for cubics and so on etc these are just examples of polynomials and they can form a family which is nicely described by this format here then let's just take a momentary look at rational functions and see what sort of family they might form the rational functions well we start with a very familiar one this one here this is the y equals 1 over X perhaps the simplest rational function and you see one of its characteristics is that it has an asymptote a vertical asymptote it also has a horizontal asymptote so it's sort of expressing the fact that rational functions may have both in fact they may have both one or either and this is different from polynomial functions which we just saw have no asymptotes here is another example of a rational function y equals one over x squared plus one it looks like this it has a horizontal asymptote as you see but no vertical asymptotes so it's not guaranteed that there'd be a vertical asymptote if you're a rational function and then etc in general you can get all sorts of pictures here's a picture you might get you might have something where you have a couple of asymptotes maybe the rational function goes up like this maybe does something like this in between and maybe this off to the right so the y axis and this line here wherever it is this vertical line is an asymptote vertical asymptotes and maybe we could even do this and get a horizontal asymptote here so rational function can have many of these properties together it really depends on exactly what its form is but these can all be lumped together sometimes when we want to talk about them as a family

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Length: 36min 29sec (2189 seconds)

Published: Tue Jun 16 2009