Videos

Function Vs. Relation

Added Jan 28, 2015 •

Copy this URL:
Embed code:	<iframe class="edlio-embed-player" type="text/html" width="640" height="360" src="//compass.jisd.org/apps/embed/?v=70854" frameborder="0" allowfullscreen></iframe> Change dimensions

This is a detailed video describing the difference between a function and a relation and how to determine whether a relation is a function.

Show Transcript

I have a screw in this video we are 2 Define relations and functions and compare them together we are going to first start off by sort of making sure you understand the basic structure of the coordinates some look at some different versions or warning that we might see in our textbooks for looking at the sort of X and Y variables if you will but battery please have an X and A Y in it they generally do for grafting them on at XY coordinate plane which am I took but we're doing a lot of emphasis on so you're going to see a lot language about just using x's and y's but Alberta Lee sort of elaborate and he's definitions that I gave you and use some of these fancy or words go to Define what a function relation is comparative comparison over on the side here we can't see yet we're going to look at the difference between a relations and functions by just simply looking at coordinates points will be doing that with the either visually or with the mapping that I have here for you already set up we're going to compare relations and functions graphically using the vertical line test which I'll explain as we go and then finding the last phase of this video if your use and maybe a more advanced book like my precalculus book does a lot of questions wear chapter look at equations and just from the equation determine if that is a a function of Y in terms of x  so jacober and hopefully in the end of all this you get it all tied together and makes perfect sense so when you look at a coordinate system specifically for graphing of course we this is a rectangular coordinate system polar coordinates is Polar Systems and serve an imaginary type system where this is a real and imaginary axis put on a regular coordinate system the next Sox's we have a y-axis we have plotting points if you're in quadrants we have quadrants 1 2 3 and 4 it starts in the upper right hand corner and then goes counterclockwise as you'll able to court the quadrants 1 2 3 and 4 and if you're in quadrant 1 both your X & Y coordinates going pawsitive if you are given a coordinate where the x is negative going to move to the left on Quadra to the ex's or negative and because you're above the x-axis the Y is going to be positive in quadrant 3 both the x and y coordinate or part of that court on that point the court of that points are going to be negative and then Quadrant 4 your ex is positive and your why is negative just to make sure that you you know understand completely the structure of your coordinate system and where the X & Y axis meet that all the origin that's .00  now if you're just doing graphing of course all the variables are going to see your ex and one but if you're dealing with word problems maybe your trying to explain how many miles or kilometers per hour well skewes me miles per gallon or the beat  liters per kilometer maybe not sure how different countries measure the efficiency of their cars but if I wanted to explain miles per gallon how far was getting on a gallon of gas then I could maybe say how fast am I driving and how is that related to the efficiency of my car the faster I drive after certain point I start pushing all that air in it and reduces the efficiency of the car so if I was doing some kind of work problem relating speed and efficiency your miles per gallon that my car was getting I would be able to control the speed as a driver so the speed would be the independent variable and the dependent variable play you know the efficiency that my car is get the chibi my miles per gallon if you will would be the dependent variable by speed would be explaining that Miles said that mileage we could be looking at just calling the X the input what numbers are you putting into an equation and the wand I use your answers as with equation play van out to you and the domain and range some vocab that you will find in every single textbook and it's a huge deal identifying domain and range solve a function or graph where the domain is going to be again the input or the independent variable or the x coordinate the horizontal movement how long to craft in the range values for your output or your dependent variable or just simply the y-coordinate and that would be the vertical axis of our XY system so lots of vocab there and if your textbook as emphasizing one version of this over the other hopefully it's doing a pretty decent job of explaining those definitions and helps a function each X has only one y now I'm starting with function cuz we kind of like questions always seems like my books are using questions like you know is this a function and it's yes or no really a function is a special type of relation and well instead of me just sort of pointing to what's out of the screen this one pause this I move the camera little bit and Define these both for you and skip to those examples  so you can see both a function each X has only one y that sort of like a very simplified where's the definition of function and is great if all your variables just or your problems involve just text him Libreria bulls or you just doing graphic but again with the word problems setting or if you just don't have some more complicated problems you might hear or read something like the function is set of all ordered pairs such that element number in the domain your input or your ex's corresponds to only one y or only dependent variable so each element of the domain corresponds to only one element of the range or independent for the Babson defended for the Y so kind of like those words I put over there about independent dependent input output domain range your ex would be your for your input or your domain value and your wine values that your dependent output or Ranch is a city where one some or all of the x's in your set of points or in your graph can have more than one wife you might see definition your textbook of relation is just simply a set of ordered pairs the other words a function is a special type of if I give you a set of ordered pairs that's a relation kind of the difference between a quadrilateral skull in square like a quadrilateral is anything with four sides a rectangle has four sides the opposite pairs are equal and all the angles are right angles and then even more Nestle the knowledge of square which has four sides but yet they're all equal and their angles are 90° so the relation all set of ordered pairs but this seems to help soon see the difference a little bit better between a function and a relation again one some or all of your exercise can have more than one wife as opposed to a function where every single value of x every single value or element your domain can only have or correspond to 1 elements in the rain for only how when why okay please represent a function yes or no well I can look at you and see well there's only one there's only one value of one it doesn't repeat again so that only has one value lights there's only one value to it doesn't repeat so there's only one point with a cordon of too and so does this too is only paired up with that wine value for and say the tree so I can just look at this and see that's a function if you're doing a lot of napping which is a very nice ideas well you can draw an ellipse and put all the values in your range for your independent variables of your ex's in one oval or lips and another sort of all the land put all your values of y and I'm up we have one in Forest let's draw a line going from 1 to 4 the x value of 1 to the Y value for and we have 3 and 6 directors values of s 1 and 2 button here we come in with the idea of the dependent or she's medium independent variable or their domain are the inputs okay whose values were getting inputted into a function and we're getting out these naughty so when you make a mapping like this make sure you keep your ex's on the left in your wife can you ride your domain in your range and you read it left to right this text only goes to one why the sex only goes to one one now it's the same line but that's not with this sort of Simplify definition says each X has only one y does not say anything about going reverse each why only has one X this is a function as opposed to over here we have 3231 and 376 now x value of 3 shows up in two coordinates which means that that x-value three is paired up with two different values of why that is relation you know all set of ordered pairs  are automatically relation so it just doesn't spit starfit the special requirement of being a function as a mapping we have the values of 3-3 and 7 now I can't you notice here I did not repeat the value of 3 I'll send over here with the wine value for if I repeat the value of 3 is I go down my list and I'm listing all my values of the domain can we do not repeat that if I did it would hide the fact in my mapping that the x value of 3 I want to see you know if you see the split in your map that tells you that value of x goes to two different values of wine and that's why that doesn't meet the recession requirements to being a function it's only a relation  whether we have the Turn 6 again if you didn't 3 - 2 + 3 - 1 you wouldn't see that for going left to right K we read these left right the X values read Independence the domain values the input values these what we're getting out these are the again dependents change for the output to Renys left to right this is a relation this Fork again going left to right is okay this is a function looks like about graphical comparison there's something called a vertical line test and a vertical line  the vertical says says that if you can I don't have this written up here sorry but the vertical line test says that if you can draw a vertical line through a graph and it holy crosses wants no matter where are you draw that vertical line it is a function well why why is this vertical line test for graphically testing whether or not something is a function well let's not forget that the equation of a vertical line is x equals enum and the function says each value of has only one want so I care about what value no do all the values of X only have or correspond to one value of y so every play draw a vertical line I'm representing a different x value so this x value has one want this x value has one wine the sex value that's one wine so on it doesn't matter where I put my vertical line it'll only ever crossed once and that means that this is a function  this example we have something called a piecewise function if you have studied those you should understand what how they work and if you have it maybe you can just skip that or look up my video piecewise functions of the piecewise function and we have a line here with a positive slope and it ends with a hole distance down to the left forever and as indicated by the arrow this whole means that let's say that this stops at value of 2 so this part of the function goes up to up to the value of 2 and when you get to two it's not included like this piece of the is only 4 when X is less than  less than 2 so there's an open. They're saying that you know the number two is number in the scrap but this particular line with the Positive slope you know goes up to like one point nananananananana but not quite too will open at 2 and then there's another piece of this piecewise function that's a flat line it's a horizontal line equation of a horizontal line is white was a number play this part of the piecewise function is only going to be good for exercise that are greater than or equal to 2 so equal to 2 so we fill in the Sala. We start that part of the piecewise function with solid. Indicating that value of 2 is for this bottom horizontal part of the function well it's troll those vertical lines process once crosses once let's see here does a Crossing once or Crossing twice there's no this part of the line does not exist there I'm just marking the end of the line at sitting on goes up to 1.9 another but not quite to the value of 2 is down here so this is only Crossing once and this is crossing once a function over here and have the sort of like sideways s shape if I can draw vertical line up only crosses once only crosses one only crosses once but if I put it down here somewhere I'm crossing that graph and does have to be three times it just has to more than once that is a relation blood of textbooks like to do a sideways problem for that as well here I have another piecewise function it looks remarkably close this one but you'll see that I have Mark two solid dots so this part of the piecewise function is for when Silas just keep the green look for that this part of the piecewise function is 4 when X is less than or equal to 2  Stalin's March now this part of the piecewise function also has a solid. So it works and we want this part of the piecewise function when X is greater than or equal to 2 and if I prompt that vertical line what represents a value of x  because those two dots are solid that would also now you know millions of points where I can draw a vertical line and have it only cross once but there's that one special line that I can draw all that one vertical line at one fax this one value of x that's paired up with two values of wine and even though this works you know millions and millions of time so we can here and here and here can your hearing over there that one time it fails what does it look like let's find out four examples determine whether which equation defines y as a function of X well Y is a function of X or Y in terms of X that language kind of gives you a hint as to what we do to determine if it's a function where each X only has one y or just a regular relation Y in terms of X or Y as a function of X implies me telling you that you need to get wine by itself and it is helpful stepper process into the determining whether these equations are relations are functions without going to see a picture we're going to subtract by 2x squared and if -1 is equal to these or not like terms so please do not do 6-2 this is going to be 6 mine say good for - 2x squared and then we're going to get rid of that negative in front of the wine by dividing everything by -1 why is equal to -6 + 2x squared know you might recognize from knowing your parent functions go all the way up to hear if you know your parent functions if you have to sort of like a regular polynomial here with just the X being raised to a power of 2 this is a problem now weathered opens up or down we're not sure until we have it Sulphur why the square turns has a positive leading coefficients this is a opening up well I just got to cheat it cuz now I know my parent function so I know roughly well at least that this is a parabola opening up and I wreck places that is being a function without that though you know when you take a number you square it like three Square does 9 not 2 answers x 2 when multiply you don't get two possible answers and subtract from 6 so there's nothing going on here that would ever have me plug-in X in and get more than one wife so this is a function  get over here so we'll just put a check mark for the functions in Annex for the ones that aren't okay here with x squared plus y squared equals 49 well I didn't say we were doing this without a ground but again if you know your basic structure of some graphs play I have two squared terms and we have them both being a therapy and positive both have the same sign so you might recognize this as being Circle this is a circle whose radius is 7 and if you happen to know what that function looks like oh well if I drop a vertical find it's going to crosswhite so this already know his a relation but how would that look how would you figure that out if you didn't recognize that as being a circle and it would be helpful could well we solve it for why like I kind of said we were do with the directions to Renaissance Signs by x squared + y squared is equal 49 Elm subtracting this positive x squared a moving it over from the left to the right hand side by doing the opposite math operation at the positives or moving it over with traction I need to solve for y and having a second power on their power of 2 is not does that mean it's not done yet so we're going to square root those sides and now I must pause to just rant a little bit to beg and plead you all in my own students I understand that 49 is perfect square that the square 249 is 7 + Annette x squared as a perfect square but this is not equal to 7 - x you cannot square root 2 returns even if there are perfect squares square root of 4 + 9 as an example is not equal to 2 + 3 this is not 5 this is a square 213 so if you're my students watching this or anybody else for that matter there's a lot more than just my students watching this tell the wrist do not swear Ruthie's when they are separate terms if this was one term if there was a multiplication in the middle and it was 49 x squared then yes would be 7x but not with the minus sign not with the plus sign not when you have multiple terms inside your square root symbol this is going to be just 49 4 - 49 - x squared some space here  and I have another question here with a square root in it or even redoes have to be read before through six through eight through when you put your own even Roots into your work I had to put that square root pause besides this equation I must account for both a positive and negative answer and film right there when I plug in an ex and I workout my stuff if it doesn't come out to B- inside here and I can actually take the square root of this value I must account for both the positive and negative answer because I introduced even rude that means that I'm going to plug in the next and get to separate wine values and that means that this I know us isn't due checking on but this is a relation okay now we're here this is done this is all for why and this is a function you said that you know square root Evenrude you have that two possible answers positive and negative but this is how I'm giving you the equation this would be maybe something that you see in your book and when the author gives you an even rude keep saying even cuz when you take the third root of a number or an odd root of a number like the the third root of 8 is just two two two and two is eight the third root of -8 is -2 negative to negative to negative 2 is negative but with even Roots the other plus or minus thing comes up it's the author gives you an even routes they're going to let you know whether they want the positive answer the negative answer or both if you don't see a plus or minus put in here then the author is only expecting one answer I have talking funny hear some expecting the positive answer so this only has them if I plug in an X and if this is all workable which it's going to be when I plug in an X and I take the square root I'm only going to have one value of y a positive value of line because there is a there's no negative indicated in front of the square root or the even resemble so this is a function  stay over here if I did my own Evenrude I must therefore count for both the answers and that was a relation we have an odd power but we're just saw this for wine we're going to do why huge we're going to add 4X to those signs and undo that subtraction so we get 8 + 4x I'm going to get rid of that power 3D inverse of the third power is the third root so we're going to put the Third all those sides and yet I might be above you know the difficulty level of your class at this point but the third root the third power to cancel out we get the third roots of 8 + 4x and that is also going to be a function because when you multiply you only get one answer when you up at 8 to something or add two numbers only get one answer when you take the on roots of a value like the 8th the -8 I just did when you take an Audrey to the value there is only ever one answer from that so this is also going to be a function well that is about as much as I can say for the difference between a relation and a function I'm looking for my remote but it's not with me today that's all I can say about relations and functions I miss true thank you very do your homework

Uploaded by Heather Avera.

Videos

Function Vs. Relation

More videos from Heather Avera