M1110 TB 2.3 Functions

and in a series of ordered pairs a function is such that the X element is only used once in the series of relations so let’s take a look at this an example one it’s to decide whether these are a function or not well let’s take a look at them so here we have a one we have a negative two and a negative three are the X elements repeated know so yes this is a function here we have a 1 111 repeated so this is not a function a negative 4 negative to a negative to the negative 2 is repeated it’s not a function now sometimes you see it shone like this where these are your X values these are your y-values now notice from the X values there’s only one line going from each value so this indicates that these items are only used once so this is a function now in this case we see from the negative 2 which is the x value two lines so that indicates that’s used twice so this is not a function now we’ll also see it in drafts from time to time as you look at that and you graph it notice the X is only used once so this is a graph of a function now a little later we’re going to show you what is called the vertical line test in fact i’ll show it to you right now now in the vertical line test we take a vertical line it’s an imaginary line that we move over the face of the graph now as we do this does this vertical line intersect our graph which are dots here in more than one place so as it goes over this one only one place it goes over this one only one place goes over this one only one place so yes this is a function now here again they’re relating a slope intercept formula to a function where your dependent variable results from putting some value for the X here as you put in some value for an X let’s say a 4 is what they’re showing then 2 times 4 is 8 and 8 is the output so we have this as the input with X the independent variable and then your Y comes out of the system now to tie this in to to graphing let’s say if we have a graph and we make a graph of this where our y-intercept is 0 we put a dot at 0 the slope is 2 over 1 so we go over 1 up to and then we make a line through that and in a sense this is the graph of this function we can change the wide f of X now does this line pass the vertical line test well as we move this line over the face of our graph does it intersect it at only one place as it moves across and the answer is yes so this is a function and that’s what they’re saying here as we go on they’re introducing domain and range which we’ve defined already the domain is the X element the range is the Y element so if you are asked to find the domain and range of a relation here are your coordinates I

suppose it is then it would be a usually they would put like a dot there to indicate that so that’s a little tricky there you would say a negative 3 if it’s included it’s not included it would be a parentis e and this goes up to positive infinity up over there okay we’re going to look and see what the book tells us here in a moment too I’m just interr so we were good this is shown in roster notation this one is shown an interval notation and these numbers are included because of the brackets infinity and infinity that was good and then use of a bracket here 3 negative 3 is included okay now in this area where they’re asking us to look at agreement on domain we think of the number line and all the numbers that are on the number line are real numbers so how about the case where if a value such as let’s say 2x plus 3 is over 2x and we’re asked to determine the domain which would be the X values of this example well we have to keep in mind here and that’s what they’re referring to an agreement on domain that we can never have a zero for a denominator because that’s not in a real number system where we have a zero for a denominator we say anything with a zero for the denominator is undefined so are there any values of x here that would make this denominator a zero and the answer is you might say well how do i find that will just equal the denominator 20 and when we do that we get x equals 0 so if x were 0 that would make this denominator a 0 so we then have a restriction to what values we have for our domain in order for it to be a real number so we say that often in set-builder notation will say X is such that X is any real number but X cannot equal 0 so here you have a qualifier to your domain and this is set builder notation now you might say how would you put that in interval notation well again I’m looking ahead maybe hopefully it’s their interval notation that would come from negative infinity up to 0 but a parentis e that 0 is not included in the set and then we use a union symbol to indicate we’re linking it to the other side where it’s going to be the other side of 0 and positive infinity so an interval notation that would be the answer to this one so this is what they’re saying agreement on domain that X could be any real number but there are sometimes exclusions as to what X could be and we would find that always by taking our denominator equal to 0 solve the equation and that gives you the values that have to be excluded and I see here they’re introducing the vertical line test okay we’re just a little early and again imagine it’s a vertical line that you are going to pass over the face of the diagram over the graph and if it only intersects in one place then it is a function now here in the case of a circle it’s intersecting there and they’re so circle is not an exact of a function and they’ll give you many

down here this gives us a sort of sideward parabola now is this a does it pass the vertical line test no and it’s not a function now the domain and range of this would be that it would start at zero for the domain and go to positive infinity and then for the range this eventually comes up from negative infinity to positive infinity ok let’s go on and again all the answers are down below so you could look at that too i’m just giving you a little interpretation of it so when you did a graph of this would there be any restrictions to the x and let’s look at what they say so as we go through these we had all of these correct and they are mentioning that the radicand cannot be less than zero so they’ve solve that the way we had and in letter C we did get the domain for the positive values and the negative values and got the graph of this it’s not a function and here they’re saying there is no restriction and it would be all real numbers for both the domain and range now here’s a case that I mentioned a little earlier that where we have something in the denominator and let me review that with you one more time if we said that we take our denominator equal it to 0 and we get x equals 1 well here if X where r 1 it would make your denominator is 0 so our value for our domain in a case like this is that it could be any real number but cannot be a positive one and let’s see how they’re showing that because i had mentioned away earlier and whether they’re going to use interval notation or not yes this is what they’re going to use so the domain would come from negative infinity up to 1 but not including one and then with Union on the other side starting at one just beyond it to positive infinity okay and the range applies the same way because in this case it’s 0 though let’s take a look at that now because this original denominator was a negative 1 then 1 is excluded so in a sense this is your vertical asymptote and then horizontally when we go looking for the other value it does it cannot be zero so our range is from negative infinity up to but does not include zero and then starting on the other side of zero going up so this one’s a little tricky so again study what they have here and you know that’s what you’ll be checked on in your practice and quiz me now keep in mind all of these fit within the definition of function and if I were you I would write them out in your class notes or your homework log or whatever so that you’ll have it as a reference for you okay we’re now going on to function notation many many of you have

had this introduced in an earlier course that we can always substitute the letter Y for a function notation symbol this is referred to as f of X now this parenthesis not mean multiplication as it usually does but it’s just part of the function notation so we can change from slope-intercept and its standard form to function notation and throughout this you’re going to be given values for this X here and that would be your independent variable that you then put into your expression and then work it through to get the output which would be a y value or the value of the f of X in that particular situation so let’s take a look at some of these and often i mentioned to my students to make a little teach art so even though they’re not showing you that technique you’d put your ex there and then just put f of X here keeping in mind that would be a Y and then they want you to deal with the tomb and then they may want you to deal with other numbers here in this case we’re just starting with a two so you just put the two there this gives you three times two is six so this would be a one I like this because it helps to organize your data and I think very useful on your blue sheets when you’re doing a test and the teacher can see specifically what you’re doing so they started us off easy and now they’re giving us some more challenging items part of it is to understand what they’re asking here they’re saying let f of X equal this function but then they’re also showing G of X is this function so here for the F function we would put the two where the X is in there and the later we’re going to put a cue wherever the X is and they show this substitution down here and then for our G function we’re going to put a plus one where the X is and then just work that through as you see here ok and then again in this case maybe a t-chart might not be appropriate but you’re going to do them as different examples so here this was the easy one this one came out as three and this one will then come out as an expression where you’re just substituting the cue for the X and then here you’re putting in where the X is the a plus one the a plus one and then just multiplying through and ending up with an answer that we can’t quite see there let’s take a look at it there we go so by the time you group like terms and everything you end up with this and you’ll be getting some practice ok now for something like this for each function fine f where the x value would be three so some of this is straightforward and we can use what they show we’re just going to take the three put it where the X is nine nine minus seven is two ok and then in the case of B we’re looking for where X is 3 and we have to look where X is 3 which is this one so why is going to be a one ah so something a little different and then here where F is three right there it’s

going to be five and then on this graph where F is three well we look let me make a little picture of it for you here there’s where it’s three we go up here to right there and what is our ordered pair there well it’s three and that’s going to be up for so our Y value is 4 so a number of interesting ways in which we use function notation where they give us part of it and we have to determine the other remember they’re giving us the independent variable we have to find the dependent variable from that information whether it’s an equation series of ordered pairs sort of like a Venn diagram and then a graph ok let’s go on see what they show us we did the first one and there the others there’s our five and there’s our four okay let’s take a look at this well here they’re just asking us to find an expression in our funk using our function symbol so the key is you solve the equation for y and then replace the y with the function symbol so this one’s pretty easy because the Y is already there you’re just going to replace that with the function symbol in letter B we have to actually solve this equation for y which we did in earlier lessons in other courses put it in slope intercept form and then substitute the Y for our function symbol ok I always get for the end of this lesson we’re seeing we want to determine whether something is increasing decreasing and whether it’s a constant function well again in our language we go from left to right generally so as we go here and this blue line it’s increasing as we go in this reddish line is constant and as we go down it’s decreasing so really good example of what’s going on here so illustrate these ideas here is increasing here it’s decreasing and here is constant now as you look at this parabola remember going from left to right it is decreasing here then is constant for a moment and then starts increasing and let’s read what they say here the concept of increasing and decreasing functions apply to intervals of the domain not to individual points all right that’s interesting and now we’re taking a look at something here determine intervals over which a function is increasing decreasing or a constant well I’ll let you look at this in this first one as we go from left to right this is coming from negative infinity so that’s decreasing up to 1 but notice that one has a circle on it so we use a parentheses and then it’s increasing from one which is a bracket to indicate one is included in the set up to notice we’re reflecting on the x-axis three and then from three again bracket because three is included in that point up to infinity which is a constant alright and on this word problem I’m

going to let you read it because the tape is running 40 minutes wow hopefully it’s helpful for you give you some background and I know you could read your textbook yourself but this might be useful all right we’ll wrap it up