hello there and welcome to our new lesson this video is for senior high school general mathematics for grade 11. prepare the following a paper and a pen for you to write your answers or solutions for the problems later on and remember you can always pause and play this video whenever necessary you can even go back or revisit the portion of this video to clarify some things for mastery purposes i hope that you are all excited for this so let's hop in this video presentation is for the first quarter module 1 of our subject general mathematics for grade 11. the topic is real life functions to be specific this is for the first lesson about real life functions what you need to know we have three main objectives for this session the first one is we are going to determine functions and relations second illustrate functions through mapping diagrams sets and graphs and finally you're going to represent real life situations using functions what's in what you see on the screen right now is a crossword puzzle exactly we have here five descriptions of different terms that is related to your junior high school mathematics these are necessary terms for us to proceed with our new lesson okay so you can pause this video and try to recall those important terms i'll give you time go ahead pause the video are you done that sounds great so let's reveal the answers so for number one let's have number one down a rule that relates values from a set of values which we call as domain to a second set of values which we call as range what do you think the answer is relation so let's put it in our crossword puzzle relation there number three three down blank pair pair of objects taken in a specific order what do you call this blank pair the answer is it's an ordered pair very good so let's put it in our crossword puzzle now to clarify about ordered pairs we have here an example remember that ordered pairs are a sequence of two elements like for this example one and two they are enclosed in a parenthesis and they are separated by a comma okay that's an ordered pair let's proceed to the next one how about across number two the set of all x or input values can you recall you have there your clues o and i for the second and the second to the last letter and the answer is domain right brilliant domain let's review about domain when we say domain look at the example we have four sets or we have four ordered pairs in this set one seven two six three five and four four now what is our domain here our domain are the first elements inside the parenthesis or first element in each of the ordered pairs so that means it's 1 2 3 and four which serves as our domain how about for number four across collection of well-defined and distinct objects called elements that share a common characteristic you have this when you're still in grade seven the answer is well done that's set s84 set last one across number five the set of all y or output values what do you call that your clue there is it ends with letter e you already have the domain this is the pair of domain that means we are referring to the range okay we have completed our crossword puzzle but before that let's clarify on range now using the same example for the domain we have here this set of ordered pairs we already have one two three four as our domain earlier right now this time the range is these values the second element of each ordered pair or the y values that will be 7 6 5 and 4 in this example what's new what makes relation a function a function is a special kind of relation because it follows an extra rule just like a relation a function is also a set of ordered pairs however take note of this every x value must be associated to only one y value i repeat every x value must be associated to only one y value that's the most important part of this lesson remember that that's for the definition of our function illustrations will help us a lot to learn functions easily so we have here mapping sets and graphing a function is a special type of relation always remember that in which each element of the domain is paired with exactly one element in the range a mapping shows how the elements are paired it's like a flowchart for a function showing the input and the output of values like this the domain for the first set and the range for the second set now in this mapping let's identify if this is a function or not a function how do we do that recall every x value must be associated to only one y value so basing on that let's try to check if every element in our domain is associated to only one value in our range let's focus on this part our domain a is associated to roman numeral one so that's one is to one that's the correspondence second domain or second element b that is associated to only one y value that is roman numeral 2. here c third element of our domain is associated to or is being paired to only one value of y that is 3 or roman numeral 3. and lastly d in our domain is being paired with roman numeral 4 in our range so as you can see every element in our domain is being paired to only one value of y in our range so that means this example is correct this is a function let's look at example number two can you identify if the given is a function or not a function you may pause this video okay all right so how about this example this is still a function y looking at all the elements of our domain negative 3 is being paired to 0 negative 1 is being paired to 4 2 is being paired to 7 and 4 is being paired to 4. so this shows that every element in our domain is being mapped or is being paired to only one value in our range which means that if we have an input of negative three the output is only zero if we have an input of negative one the output is only four we don't have any other y values if we have an input of two therefore our output is seven if our input is four our output is also four this type of correspondence shows many is the one for this part we have two elements in our domain here that's negative one and four we have two elements in our domain that has the same value in our range take note what we are referring to in a function is we have every element in our domain is paired with one element in our range which means that for every input there's only one output this type of correspondence is considered as a function i hope that's clear so this is a function third example how about this is this a function or not a function you may pause this video and let's reveal this is not a function why earlier we saw many is the one correspondence right this time you call this type of correspondence recall your grade seven and grade eight mathematics in your junior high school this is for this element in our domain which is letter a it's being paired let's focus on that here that's our domain a it's being paired to one roman numeral 1 in the range at the same time the same element in the domain is being paired to roman numeral 3. now that means this element in the domain has two outputs one and three which is clearly a violation of the definition of our function right therefore basing on that element this example is not a function i hope i made that clear a is being paired to two values in our range we are done with the mapping again these are illustrations to help us out understand or identify if the given is a function so this time let's move on to sets sets in this example we will have rooster notation so we have a set of four ordered pairs beginning with two three four five five six and we have six seven now can you identify if this given set is a function or not a function now how do we do that let's identify first the x and the y elements in each ordered pair so for the first one here two is our x sub one three is our y sub one four is our second value of x in the second ordered pair five is our y sub two five is our x sub three in the third ordered pair six is our y sub three here in the fourth ordered pair this will serve as our x of four and this will be our y sub four which is seven now why is it important for us to identify our x and y's in each of the ordered pairs because these values in a domain are the critical values so identify if it's a function or not a function why look at this we have 2 4 5 and 6 in the domain no x value is repeated so 2 is distinct from the rest of the domain that's 4 5 and 6. thus we consider this as a function this set is a function remember that when there's no x value that has been repeated in the given set then that means it's a function second example this set we have three three four five five five and five four so again the first step is let's identify the x and the y elements like this followed by yes we are going to identify the domain so meaning all the x values in our ordered pairs we have 3 4 5 and 5. now notice that here 5 is repeated that's the x value it's repeated for that element in our x or in our domain we do have two different outputs which is not anymore the definition of a function so this is not a function well done we are done with the second illustration for sets again we are done with mapping and we are done with sets now this time let's focus into another way that's for graphing how do we identify if the given graph is a function or not your clue there is being pointed it's vlt that would be our magic keyword to identify if the graph is a function or not how what do you mean by vlt vlt stands for vertical line test yes functions can also be determined in graphing we can use this vertical line test which is a special kind of test using imaginary vertical lines and to check if these vertical lines would touch the graph only once otherwise it is not a function what do i mean by that if the vertical line hits two or more points on the graph it is not considered a function let's look at some examples look at this graph how would we know if this graph is a function or not again what's our magic keyword we'll be using vlt that's the vertical line test right so that's it the blue line that you see on the screen right now that's an imaginary line yeah it's not part of the graph we just made that line to test if the given graph is a function or not i hope you're following so the point there is here which means that the line the vertical line touches the graph at that point only once now let's move the blue line let's move the blue vertical line because here you can check if it's a function if any point of the graph it would only touch the graph or the given graph once so let's move the vertical line how about there yes it only hits or it only touches the graph once how about there only once and finally right here yes it only touches the graph once hence we can say that the given graph is a function so that's an example of a function basing on the graph let's look at another example identify if this graph is a function or not a function i'll give you time you may pause this video to give yourself more time all right are you done let's check let's create our imaginary vertical line again we will be using vertical line test right there the black dot represents the point where in the vertical line touches your given wrath it's only once right let's move it a little bit to the right right there it touches the graph how many times still once let's move it there still once last one over there still it touches the graph once basing on that we can conclude that the given graph is indeed a function so that's an example of a function now let's practice more let's identify if these given graphs are functions or not a function again let's identify try these graphs you can pause this video right now and give yourself more time to scrutinize each of the graph and identify if it's a function or not a function go ahead [Music] okay so let's reveal the answers now based on the sixth graph we can actually create two groups and the first group consists of these two graphs now let's focus on this point right here for the first graph as you can see the vertical line touches the given graph once how about for the second graph there it only touches yes it touches the given graph same with the first graph only once let's try to move the vertical line to the right right there it touches still the same once let's move it more right there still touch us once and last one same result once thus we can conclude that these two graphs are considered as very good we consider this as functions so the remaining four graphs looks like this observe for the first graph we have here two points which means that the vertical line touches the graph or touches the given graph at two points however for the second graph this would be the first one and that would be the second one still the same it touches the given graph twice the vertical line touches the graph here at the same time here so that means there would be two points right and lastly we have here the last graph it touches the graph twice let's move the vertical line like this well observe that for the fourth graph you now have three points earlier it was only two this time as we move the vertical line it touches the graph at three points now for the first three graphs it's the same it touches the given graph twice let's move it there observe that in all these given graphs the vertical line touches the given graph more than once again that's more than once because for the first second and third graph it touches the graph twice for the fourth graph it touched us earlier the graph twice this time thrice it's more than once yes which makes all of these graphs not a function so these are examples of not a function so those are the illustrations again for the mapping sets and graph how about functions in real life this is a circle so an example of a function in real life is the circumference of a circle the circumference of a circle is a function of its diameter it can be represented as circumference or c of d is equal to d pi alternatively we can also use it as a function of radius which is c of r is equal to two pi r [Music] another example is a shadow the length of a person's shadow along the floor is a function of their height and the third example is driving a car when driving a car your location is a function of time what's more i prepared here a 10 item assessment first to check your understanding for our lesson for today let's try to have a closer look you can pause this video or you can even take a screenshot and answer it later during your available time so we have your items one two and three again you may pause or take a screenshot [Music] okay let's move on the next set is for items four to six next we have seven to nine again you may take a screenshot or pause this video and finally we have item number 10. [Music] if you're using the same mode you'll do not forget to submit your answers to your teacher on your agreed date and time [Music] what you need to remember a relation is a function when every x value is associated to only one y value do not forget that you can illustrate functions through graphing mapping or sets and lastly functions can be seen in our daily lives like driving a car wherein your location is a function of time the length of your shadow which is a function of one's height and a lot more and that's it we are done with the first lesson for this topic functions for our general mathematics subject great job for today see you in the next lesson