okay uh good morning everyone so for today's session we are going to discuss the continuation of last week's discussion regarding the functions as models so last time we've discussed the five models the table of values mapping diagram equation the graph and the machine so we've also discussed the differences between relation and function okay so for today we are going to discuss the domain uh real-life examples of function and at this point when we are evaluating functions okay the last time nanak discus style is all about the functions as an equation where in remember domain is a set of values that the variable x can take okay so simply it is the set of values or possible values of x na predicting the meeting possible values x almost satisfies given functions or equations okay and uh we will use these five examples of functions as an equation so for letters a and b now prove nothing that all real numbers will satisfy in our given function 2x plus 1 and x squared minus 2x plus 2 meaning fractions decimals whole numbers integers natural numbers we can we can use all the real numbers and maxes out is five padinyans function okay for a letter c uh although this is not a function pero di because it will result to a certain value of y pero again please take note that letter c is not a function because this is an equation for a circle okay for a letter d we prove that we can use negative one and all positive numbers of the positive infinity so in case negative numbers it will result to an imaginary number take note an imaginary number does not exist [Music] this is a fraction type so we can use all positive and negative numbers up to positive or negative infinity except positive one so we cannot use positive one because it will result an undefined number okay let's see x is one one minus one that is zero any fraction number zero as denominator it is called an undefined number and then uh we also have this one so this is what we call the function of x this is a line between why f of x q of x g of x r of x so meaning we are going to use the value of x for us to find the value of y and um without the value of x we cannot uh determine the value of y we cannot calculate the value for y that is why uh nereida please netanyahu f of x q of x g of x r of x where in uf okay next we are going to discuss functions as representations of free life's equations [Music] [Music] scientists mathematicians functions and when we say functions involve d two four fundamental operations okay so they are called addition subtraction multiplication and division okay so we'll discuss number one give a function c that can represent the cost of buying x meals if one meal costs 40 pesos so i think an example can simply impact billing pero maron time given the cost of buying x meals so you are quantity i hindi indicates a problem nathan instead we use the variable x and then cos that is 40 pesos total payment for a certain number of meals and a gamete nothing operation i multiplication therefore our function would be c of x is equal to 40 times x or 40 x meaning we are just going to get the product of 40 which is the cost and the x nation which is the number of meals for us to know the total cost of buying those meals okay for example x nathan is equivalent to five so bhumilitaine and five meals that costs uh 40 pesos each song is 40 multiplied by 5 and we'll be having 200 pesos okay so on total cost and when we buy uh five meals is 200 pesos we cannot use division 40 divided by 5 is eight pesos so very impossible cost per each sa total cost another one we can use addition okay okay substitute okay so that's for example number one and this is a real life application of functions another example 100 meters of fencing is available to enclose a rectangular area next to a river give a function a that can represent the area that can be enclosed in terms of x 100 [Music] so let's see this is our illustration this is the rectangular field and uh besides the upper part naimai river so your 100 meters fencing material sides okay so perimeter it is uh the sum of the sides now is the function of the area remember the formula of having an area of a rectangle is length times width and the formula for having perimeter is to length plus to win okay again this is the function itself land times swede maritime conditions a problem is in terms of x so instead of using l and w we are going to use x and y okay so an area nothing is gigging x times y a is equal to x times y where in an x naught n is the length and your y naught n is the width okay along with the starting problem [Music] formula or function for the area okay so again [Music] okay this is one hint perimeter and with the use of the perimeter area okay so let us substitute the given value the perimeter is 100 meters is equal to two length plus two width kayalang if we're going to base our uh equation little satin problem and our illustration hindi unattended fencing material at an upper part river that's all right you 100 meters so definitely if x is our length okay plus so we are going to write it as one hundred is equal to x plus two y and this equation happened even hundreds with the use of the formula given satin perimeter predict okay so if we're given a is equal to x times y y muna ang unanglaten we can formulate a function for the area okay so this will be our goals and we are going to use the formula for the perimeter 100 is equal to x plus 2 y interchange [Music] then all constant numbers are located at the right part and then at this point since so we need so we'll be having 2y is equal to 100 when we transpose positive x to the right part it will become negative x okay and then again so why language so we need to eliminate two by dividing both sides by two actually you can use the multiplication property multiply both sides by one half [Music] so cancel this one and then my e1 and y copy one hundred minus x over two alumni one hundred and two divisible by two parenthesis because 100 and eggs are separated by a negative sign this is considered as one item in chapati cancelled by two and at this point in an athenian is simplified since makahi will a naive 100 and x at opening anatomy cancel 100 divided by 2 that's 50 minus x over 2 instead of fraction and i'm so 0.5 and then copy x and this will be our final value for y fifty minus zero point five x so you why not then i transformed an attention to x [Music] x times y and then you x naught and copy as is and then substitute the value of y in an angling dito which is 50 minus 0.5 x okay please enclose into our parenthesis okay and then by distributing property we can simplify this x times 50 that's 50x x times negative 0.5 x that's negative 0.5 x squared please take note of the loss of exponents discussions so our final answer here is the function of area is equal to 50x minus 0.5 x squared at any final 50x minus 0.5 x squared once now given your value x starting problem at synaptic okay and that is for example number two okay next we have the piecewise functions so by definition it is a function that is defined by different formulas or functions for each given interval piecewise function is not familiar to us even ago actually india familiar gen until i taught and handled general mathematics way back 2017 so wise okay although hindi show familiar satin everyday okay for every given interval meron for example um given interval nothing is all about distance so for example from your home to your school that's five kilometers from the school to the market that's another five kilometers so this is the market at the interval net engine so this is the first interval this is the second interval so it is from your home to the school is 15 pesos destination market and destination additional for example this is an additional five kilometers and uh we are about to pay one peso per kilometer 20 pesos 15 pesos plus kilometers 15 tapos plus five and total fermo from your home to the market is 20 okay so that's the use of the piecewise function at 15 pesos because that's the minimum fare okay so take note of uh the piecewise function another example a user is charged 300 pesos monthly for a particular mobile plan which includes 100 free text messages messages in excess of 100 are charged one peso each represent the amount a consumer pays each month as a function of the number of message m sent in a month first is to analyze the problem do not rely too much on the numerical data terms okay so this is a very good example of piecewise for those uh nagumagamet yeah this is one example of piecewise internet connection continues that's good for uh three days so by thursday mahakatyam load mo friday saturday sunday s we are about to pay 300 pesos for every 100 text messages and once in a exit diana 100 interval if m is greater than zero but less than one hundred epic sub hen from one to one hundred the text message is fixed and that is 300 pesos so this will be the function one okay okay you know certain plan okay and for our second interval kapagang m nathan is greater than one hundred so one sonic exceed canadians that is one peso each okay for example [Music] um five messages so instead of 100 in the game it means 105 wi-fi mobile data is good for three days five gigabytes okay so that's the main difference of having a mobile plan at kapangikawai regularly okay and this is a very good example of piecewise function so to sum up our discussion we've uh discussed the domain for each function and we've also discussed the real life application of functions and one example then is the piecewise function you