N Gen Math 8 Unit 6 Lesson 2 Features of Functions

hello and welcome to another engine math 8 lesson by emathinstruction my name is Kirk Weiler and today we're going to be doing unit 6 lesson 2 on features of functions now last time more than the last lesson we looked at what function was right and we saw that functions could be represented in all sorts of forms equations graphs tables ordered pairs today we're going to be working mostly with graphs as we talk about different features of functions things that functions have about them so let's jump right into that in the first exercise now recall the functions are simply rules that it's not assign exactly one output for given input functions have some specialized terminology associated with them so let's begin by looking at some of that terminology right beginning with what we call the domain and range let me bring this up little bit so that you can see it better the domain of function is simply the set of all input values to function and you can oftentimes think of those as just the x-values so all of the inputs are what are known as the domain of the function now the range of function is the set of all outputs of the function one thing that's little bit dangerous is that the word range gets used lot in math it's used in statistics it's used in function analysis it's used in lot of different areas all right but in the area of functions the domain or all the x-values all the inputs and the range is all the outputs all the y-values so let's take look at that definition and using that definition in exercise number one function is designed to define by the set of ordered pairs shown below zero five two seven three eleven seven four and ten five letter list the domain and range below do not repeat values all right so let's talk about that little bit again the domain is all possible inputs to the function right and that's pretty easy here zero two three seven and ten right those are the inputs the inputs are zero two three seven and ten right if if you gave me different input like equals four I'd say well look can't give you an output right because four is not in the domain of this function now the range the range are the outputs themselves so it's simple enough right I'm gonna just start listing the y-values the outputs are five seven eleven four and five but I'm not gonna list the five again mr. Weiler well we get points marked off if we list the five again well don't know that's up to your teacher but you shouldn't okay the idea just simply is well what values of come out of this function well the values for five seven and eleven come out do they have to be in order no they don't set is just collection of stuff so there's absolutely no need for the range in this case to be listed in any kind of numerical order but if you will it's sort of bad form to list the five twice we don't need to write all this is telling us is that the outputs are 5 7 11 and 4 now let her be explain why adding the ordered pair 2 comma 1 to this set would mean that it is no longer function alright this is really really important question because it goes at the heart of what function is so imagine for moment that kind of scratched this out put comma 2 comma 1 why would it now be problematic to claim that this is function pause the video now and see if you can figure that out well let's go back to what function is function is rule given in lots of different forms including sets of ordered pairs that for given input you get only one output for given value of there is only one value of that comes along with it but see that's why this point is so problematic because we also have this point so because of this point this point cannot be allowed due to the fact that an input of 2 right would now have 2 values and our sorry 2 outputs equals 7 and equals 1 therefore it would no longer fit the definition of function because the definition of function is for any given value there can be only one value that matches up with it so my my explanation would be the input equals 2 would have two outputs and that's probably good enough explanation but I'm gonna list those two outputs equals 7 and equals 1 all right very very important oftentimes when you're tested on functions this idea comes up this idea that look you cannot have inputs that repeat let me say that again inputs can't repeat values can't show up more than once values can right in fact in this table get rid of this are not table but in this collection of ordered pairs value of 5 repeats but that's ok because an value of 0 only has one value that goes with it an value of 10 only has one value that goes with it there's nothing wrong with that from functions perspective right what's problematic is all of sudden if say here's you know here's an input what are the outputs can you imagine if like tossed ball into the air right and and said hey how far above the ground is it when when like like 2 seconds after toss the ball in the air and you said well the ball is at 15 feet above the ground and 22 feet above the ground somebody would be like wait no like the ball can't be at both of those spots for that input there's got to be only one output when put in you know 3 seconds after toss the ball in the air ok let's keep going and talk about other features of function now most of the other features the functions that we're going to talk about really our best thought about in terms of the graphs we could still look at them in terms of ordered pairs and tables and all of that but graphs are particularly helpful and again remember with the graphs the x-values are the input the values the values are the function values themselves they're the output right so let's take look at that in exercise number two the graph below shows is function of answer the following question what is the maximum value the function reaches all right so let's talk about this phrase the maximum value the function reaches okay the values of the function are the outputs they're the output so with that throwing ball in the air they're the heights of the ball right so if see something like what is the maximum value of the function I'm looking for the largest value the largest value and of course we can figure that out very easily looking at graph by just saying hey what's the highest point on the graph that's right there right and that's at y-value of eight right so the maximum value this function reaches is eight but of course we have an Associated question with that take look at letter at what input does it reach this maximum value right in that case what we're looking for is the value that goes with that value or possibly values now in this case there is only one peak right there and it occurs at an value believe of seven right now again those are two different things like would never want to say that the maximum value was 7 comma 8 because 7 comma 8 isn't value it's set of ordered pairs the largest value on this function is 8 and again imagine that we sort of had that graph of ball going up and coming back down right saying what's the maximum what's the functions maximum value would just be like hey what's the greatest height that the ball reaches then question would be hey what time does it reach that value right something like that now of course if we can ask about the maximum value of the function we can also ask about the minimum value of the function and at what endpoint does it reach the minimum so I'd like you to pause the video now and figure out what the minimum value of the function is and the input at which it reaches that minimum pause the video now and see what you get all right well the minimum is literally the lowest value it's down here right and that minimum value is going to be equals negative 4 and what value does it reach it at it looks like it reaches that and equals 1 all right simple enough the last question letter is the value 10 in the range of this function explain well pause the video now take look at the front side of the sheet and think about what the range is and whether or not 10 would be in the range pause the video now well the range let's write this down real quick write the range is all the outputs which are the values now we know that the smallest value right the smallest value is negative 4 and the largest value is 8 so is 10 in the range no because it doesn't fall between negative 4 and 8 no because it doesn't fall between negative 4 and 8 right every other value decimals and irrational numbers every other value we will hit between negative four and eight somewhere right somewhere like for instance we hit y-value of four right here with an value of five we hit y-value of one right here at an value of zero right we hit y-value of five right here so every other output is hit between negative four and eight but 10 is too big right the maximum value is eight there's no way we're ever gonna reach ten all right so maximum and minimum values of functions and what inputs they're hit at cuz you know you might want to know that right if you had some kind of graph that was you know showing you like the the price let's say of some kind of stock or something like that on the stock market probably going down but whatever you know you might want to know what is the maximum value the stock was at and what day did it hit it alright let's take look at exercise three and another graph using the same function graph actually it's the same graph using the same function graph as an exercise number to answer the following questions when deciding whether function is increasing or decreasing always read the graph from left to right all right so let's let's talk about this little bit right so we always want to read graph from left to right like we're reading sheet of paper and the reason we want to do that is that as we look at graph moving from left to right the value the input to the function is getting larger right as you move from left to right you are always getting larger in terms of the question is is the output getting larger in which case the function is increasing or is the output getting smaller in which case the input is decreasing so let's take look at that in letter there are four distinct portions of this function for each interval below state whether the function is increasing or decreasing alright so let's take look from equals negative 4 to equals negative 2 let's find those two points on the graph right equals negative 4 is this point equals negative 2 is this point now if were to travel along this graph from this point to this point and would have to go from this one to this one because need to go from left to right need to always be moving this direction then of course what see is that the values are getting larger right as move from left to right the values are getting larger right I'm going uphill and therefore the function is increasing from negative 4 to negative 2 now let's also take look at it from negative 2 to positive 1 and again these are always the input values so we already have where negative 2 is there's positive 1 right and of course if go from negative 2 to positive 1 then I'm going downhill on this function so the function is decreasing right going uphill then downhill why don't you finish the last 2 all right so not surprisingly here's equals 1 here's equals 7 and between that stretch we have this nice straight line on our function the function is going up so it's increasing and then from 7 to 10 the function is going down so it's decreasing probably not huge shock that it alternates increasing decreasing increasing decreasing you could have scenario where function kind of increases pauses for minute then increases again pause this for minute then increase this again etc all right so you can have increasing to increasing to increasing to increasing and likewise with the decreasing but in this case we're increasing then decreasing then increasing and then decreasing again all right now very important are those transition so let's take look at letter the location where function changes direction so when it goes from increasing to decreasing or decreasing it to increasing is known as turning point state the coordinates of all turning points now turning point is also known as vertex of function all right and they are really simple to spot real easy right here's turning point right here at the coordinate negative to positive five right turning point is literally where functions inputs go from increasing to decreasing and from decreasing to increasing etc so why don't you list the other turning points of this function all right so here's another one down here that's at the point 1 comma negative 4 and here's another one up here and that's at the point 7 comma there's hardly anything that is easier to spot on the graph of function than its turning points once gave graph of function to my daughter who was in fifth grade fifth grade and said where do you think the turning points are and she was like there they're there you know what mean the the name itself kind of implies what we're looking for in terms of turning points but the technical definition is it's where function changes from increasing to decreasing that would be one of these or decreasing to increasing that would be one of these let's keep going one more problem let's take look exercise number four given the graph of the function below answer the following questions now think you can actually answer all of these every single one of these is something that we've covered so far in this lesson so what I'd like you to do is pause the video and try to answer and take you know don't know five minutes or so should be able to get all of them at that point and then we'll go through them all right first thing they asked us to do is the minimum and the maximum values of the function remember those are the values not the values so that would be simple here's the smallest value here is the largest value now you never really to that right so the minimum y-value is going to be the lowest y-value on the graph and that's at negative 8 and in fact it's really kind of helpful to do with little subscript min min is negative 8 all right and then let's figure out what max is right that's going to be the absolute peak point it's right here that appears to be at y-value of 6 so my maximum y-value is 6 all right simple enough letter asks us from equals negative 7 to equals negative 3 is the function increasing or decreasing great well in the last time when we did increasing and decreasing it was kind of easy because did the first two for you and then it just kind of kept alternating from there on out but here you know you got to make your own call so here's equals negative 7 here's equals negative 3 well sorry equal that's not right read that graph just little bit too quick my apologies here's equals negative 7 here's equals negative 3 now of course whether you're going uphill or downhill between these two x-values depends on which direction you're kind of moving it so make sure as always to be reading this graph from left to right in other words start at the smaller x-value move to the larger x-value and what we see is in this stretch the function is going downhill so the function is decreasing on that interval all right and finally we want the coordinates of all turning points that's easy enough I've already circled two of the turning points let me circle the other two turning points right there's total of 4 turning points here they have coordinates of negative 7 positive 4 negative 3 negative 4 let's see this one is at positive 3 positive 6 and finally we have one at positive 7 negative 8 and that's it right let's wrap this up let's talk about what we saw today so functions have lots of features there's even some more that we're gonna see and follow-up lessons but we saw some of the very important ones today for instance right every function has set of inputs that's called the domain and set of outputs that come from those inputs that's called the range those are easy enough to identify right but what we also saw were things like whether or not function was increasing or decreasing on particular interval of values an interval of inputs and also right we saw the maximum and the minimum values of function there most of the time functions have largest output and smallest output not always you could certainly have like function that's straight line that kind of goes forever in two directions and maybe the minimum is negative infinity and the maximum is positive infinity in which case we'd say there is no minimum maximum all right so all of these are very important along with the very critical idea of turning point of function where function goes from increasing to decreasing or vice versa again we'll play around with lot more of these features as we move forward in our unit on functions for now just want to thank you for joining me for another engine math 8 lesson by emathinstruction my name is Kirk Weiler and until next time keep thinking and keep solving problems