one point seven functions our goal for today is to determine whether a relation is a function a relation being a set of ordered pairs and the word function will more clearly define later on and then we’ll find function values so whenever we’re given the value for any given function we plug it in using substitution and then solve for whatever that function would equal when given the value and this will be our last example so some notes what is a well what is a function a function is a relation in which each element of the domain is paired with exactly one element of the range so this down here is an example of a function each element of the domain negative three zero two and four have exactly one range or one element of the range so negative three is with five zero three two two and four negative one notice that you don’t see anything like this four gives you negative one and four gives you three for example that would not be each element in the domain in this case for giving you exactly one element in the range this case four is giving you two elements three and negative one another way to test is using the vertical line test which we’ll go over later on that would be when you’re given a graph so you’ll see more description on how to handle a graph and knowing whether it’s a function or not later on so vertical line test function or not a function in this case to test whether any given graph is a function or not we could just draw a vertical line and if that vertical line at any point touches two or more points on the line in this case the area in blue or the line in blue then it is not a function there is no place to which we could draw a vertical line on this coordinate plane that would create two touching points on the line so notice here yes we are going to have it right here we’re gonna have two spots in which our graph touches the vertical line so this is not a function in this one there’s no place that we can draw a vertical line that there would be two points touching that vertical line so yes this is a function because only one spot touches the vertical line and you could have placed it anywhere this is not a function because it is touching twice on our vertical line this is a function this little empty circle or non filled in circle is signifying that there isn’t a point here on this line it starts just immediately after so there isn’t this would indeed be a function also okay so here we’re just going to go through two examples example 1 a and B to know whether or not each of these is a function or not and we’ll explain why so when we’re looking at this this is called mapping we’d look to see does each element in the domain yield one unique or one element of the range so negative two gives us negative three and that’s it so we’re good 0 gives us 6 and that’s it so we’re good 3 gives us 6 and that’s it so we’re good and 4 gives us 9 and that’s it so we’re good a question that may arise is well 0 gives us 6 and 3 gives us 6 that’s fine where we would have an issue is if 0 gave us both 6 and 9 so as long as the domains are only yielding one unique range then we’re fine here we’re gonna see an example where it does not work so well let’s go back here yes this is a function and our explanation would be because each element of the domain yields exactly one element of the range B we have one giving us four three giving us two five giving us four and one giving us negative four the error or the issue is right here in one situation our domain or our x value or range Y our X

our input is one it yields a for our input is one it yields a negative four so this is not a function because you have an element of the domain or the x value one yielding two different ranges or two different elements in the range four and negative four you can’t have that in order to have a function if you graph this it would have it would fail the vertical line test also ok de street versus continuous functions so continuous functions are ones that you’ve seen before that would be something like let’s say a straight line or you could even have a line with some sort of curve in it so something like this would be considered a function so about both this black and this red line curved and straight would be considered a continuous function discrete functions are when you have points plotted but it’s not continuous in other words the points do there’s no lines in between them so this is called a discrete function this is called a continuous function so you’ll you’ll find that we’re going to encounter both of those throughout this year okay example two there are four parts to it a B C and D so let’s walk through them together in order to solve this problem at an ice sculpting competition each sculptures height was measured to make sure that it was within the regulated height range of 0 to 6 feet the measurements were as follows team one four feet team two four and a half feet team three three point two team four five point one and team five four point eight feet make a table of values showing the relation between the ice sculpting team and the height of their sculpture so this is just what we underlined in blue together so team one had in four feet team two had four point five team three three point two team four five point one in team five four point eight so here all we did was make a table now I put this on your notes and I also put it on the screen if you were making the table without this template you would just want to see what the relationship is between in this case it is the team or the sculpting team number and the height so that’s how I knew to come up with these two identifiers okay so now we have all of this data filled in the team number and the height and we’re trying to determine which one is the domain and the range of the function so you may recall domain is the same as X it’s the same as input it’s the same as independent variable so I need to figure out what’s my independent variable or my watt or my X and then range is my Y or my dependent variable so which one influences which or which ones dependent upon the other so we could ask yourself is the team number dependent upon the height or is the height what is dependent upon the team well it looks like the team number is independent of the height but the height is what’s associated to the team number so this would be the most accurate way to put our domain and range so once you’ve identified what’s independent dependent then you can just list out your X values and your Y values so I’ll put it in blue or domain so the set 1 2 3 4 and 5 and then in green we have our range which is four four and a half three point two five point one and four point eight okay so these are our two sets one is

the range the other is our domain okay last one Part C and D it says write the data as a set of ordered pairs then graph the data so let’s go ahead and plot the points and then we’ll talk whether or not this needs to be continuous or discrete so one four two four and a half three and then three point two four and five point one and then five and four point eight okay so now we need to determine whether this is discrete or continuous what I want to draw a line in between these two are all four of these five of these points or are they good just the way they are I would not need to connect them in this case it is discrete we would not want to draw lines connecting these points because each of them are unique points associated with the team if I draw a line I’m saying that there is a team one and a half a team one point for Team one point five or team two point eight there are none of those teams so I would not draw lines in between these points so again this is for Part C and then Part D this is discrete for example three we’re given a function negative 3x plus y equals 8 or we could call it an equation which will talk about the interchangeability of these two in a second but it says determine whether this is a function so we know functions each x-value has to have one unique Y value and we also know that if we were to graph it which we will do with this problem it should give us some continuous function and in this case it would be continuous function where a vertical line would not intersect twice within that function or intersect twice on the line so let’s go ahead and come up with some values to substitute in let’s go with negative 2 negative 1 0 & 1 so I’m gonna go ahead when I substitute these do the multiplication portion and then I’ll write it out below and then we’ll solve the equation so I chose these inputs at random I could have chosen any X values I would have liked I chose the easiest ones for this problem that will also fit on our graph and you’ll see that in a moment so again you could have chosen there’s an infinite number of X values it could have chosen none of which would be incorrect so now we’re gonna plug them in for X solve for y and that will give us our output so input negative 2 output we’re trying to determine by plugging it in so negative 2 inside of here negative 2 times negative 3 would just be 6 plus y equals 8 from here I’m going to subtract 6 from both sides and I’d have y equals 2 so when X is negative 2 y is 2 I plug in negative 1 negative 3 times negative 1 would just be 3 so I have 3 plus y equals 8 subtract 3 from both sides and I find that y equals 5 and even easier one I plug in zero negative 3 times zero zero plus y equals 8 well that’s just y equals 8 and our last one we plug in 1 1 times negative 3 here we have negative 3 plus y equals 8 we add 3 to both sides and we find that y equals 11 okay now we’re going to go ahead and graph it over here and connect the points it’s gonna be linear and then once I’ve done so I’ll then be able to determine whether it’s a function or not so here’s my ordered pair negative 2 2 so I go over negative 2 up to put a

point the next one is negative 1 5 negative 1 up 5 my next one is 0 8 so 0 8 the last one’s a good ways off the graph I’ll approximate given our graph doesn’t go that high so around in here ok now I can draw a line with arrows on both ends that goes through all of these points this line represents this function or this equation up here now I gotta ask whether it’s a function well could I draw a vertical line through this in such a way that it would touch two points on the line no so therefore it is a function yes it is a function okay equation versus function notation you’re gonna see this very very often and you’ll be writing in it very often this year and I want to make this crystal clear today so that way you don’t have any misinterpretations later on f of X is the same thing as Y it’s the same thing so y equals 3x minus 8 and f of X equals 3x minus 8 these are equivalent and you could with function notation you could have something like G of X I could have called it that same exact thing no difference I could have called it a of X same exact thing no difference a of X would be the same or playing the same role as Y now why do I put these letters out front these are just the name of the function this would be function f function G function a this would be bits of review from last year’s pre-algebra course if you’ve not taken that course it would be good of you to take ample notes on this and if you have questions address them to me tomorrow so when you are given something like 4 f of X equals negative 4x plus 7 find each value when they say f of 2 that means that we’re going to go ahead and solve this function f when X is 2 notice the X here is replaced by a 2 so when I go to solve this function wherever there was an X I’m going to put in a 2 into that function so I’ve got f of 2 equals negative 4 times 2 plus 7 so order of operations negative 4 times 2 negative 8 plus 7 so f of 2 equals this and then bring these together and I’d find that F of 2 equals negative 1 so this is my answer some questions that may come up if you either forget or were not ever taught function notation this does not mean this here F of 2 does not mean F times 2 please eliminate that from your thoughts as that is not what this is representing this is just the name of the function and we’re saying that function f when X is 2 equals negative 1 so I’ll try to read it like that for this one so it makes a little bit more sense we’re gonna find the function f when X is negative 3 and then add 1 to it so let’s do this first function f when X is negative 3 and then it’s very end we’ll add 1 so f of negative 3 that’s negative 4 substitute in my negative 3 for X plus 7 okay order of operations I multiply here two negatives give us a positive so I have 12 plus 7 equals F of negative 3 I continue on F of negative 3 equals 19

now this is we’re done for this portion of the problem right now however it says take F of negative 3 and add 1 to it so we found what F of negative 3 is 19 and we’re going to add 1 to it so 20 is our final answer okay last problem it says H of T equals negative 16t squared plus 68 T plus to find each value so here we’re taking the function H which is this right here this whole thing and we’re going to substitute in for T with 4 so H of 4 equals negative 16 and then notice there was a T here so I’m putting in a 4 now 4 squared plus 68 times there was a T now I’m putting in a 4 plus 2 so now I go through order of operations tell me that I need to do this exponent first so 4 squared is 16 so I’m negative 16 times 16 plus 68 times four plus two so now we’re working through we’re going to multiply and then at last step we’re going to add so negative 16 times 16 it’s going to give us negative 256 plus 68 times four is going to give us two seventy two plus two we add these all together and we’d find that H of 4 equals 18 and this is our answer it’d be best to state your answers exactly like this H of 4 equals 18 you wouldn’t want to just put 18 we’d want to indicate that it’s the function H when it’s when t equals 4 that’s equal to 18 okay so this one’s an interesting one it’s saying so we have to this is sort this is the order of operations so H of G means that wherever you saw T instead of T you will now use a G just like we did for back here so essentially instead of it saying T squared and 68 times T is just gonna say G squared and 68 G then it says whatever you found this to be which is pretty simple replacing T with G times that bite 2 so first let’s just do H of G so H of G is just a matter of replacing everything that says T and putting in a Jeep so instead of T squared we’re gonna have G squared and instead of 68 times T we’re gonna have 68 times G and then our two stays now we’re gonna take n times this whole thing by 2 and this is going to be through the distributive property so this whole quantity times 2 so this is where I’m going to distribute to each term inside of my parentheses so 2 times negative 16 G squared negative 32 G squared plus 2 times 68g 136 G 2 times 2 4 and this is our final answer you