hello 3us today's lesson is 1.2 functions and function notation a function is looking at the relationship between a domain and a range so your first lesson we spoke about domain and range and we talked about the fact the domain is related to your x values and the range is related to your y values a relation can be represented as an ordered a list of ordered pairs a mapping a table a graph and an equation we're going to examine a few of those in our examples a function occurs when each element in the domain maps to a single element in the range so that means that the domain values must be unique the range values so your y values can repeat but the domain values cannot so given any x value there is only one y that goes with it you're looking at a graph and i've done a little example at the bottom if you're looking at a graph a little star here must have graph okay you can use this test called the vertical line test and it is a vertical line so i've drawn you can see it here i'm kind of moving it a vertical line and how it works is i'm starting at the very left of the graph i move my line from left to right if i ever hit two points at once then i do not have a function because that would mean that my input would not be unique and what i would say is function and it passes the vlt vertical line test let's look at some examples first one is a list of ordered pairs so i have three ordered pairs if i'm going to test whether or not this is a function what i'm looking at are my input values or my x's are they all unique yes they are so i say yes it's a function and i have unique x's part b i've got 1 1 and 4 as my input it is not a function because my ones repeat okay my reasoning is repeated x values looking at my mapping diagram how this works if i was to write out the list of ordered pairs i would have one two this is my input kind of like my x's my output like my y's so i would have one two and then i would have one five and then six and nine eight and nine nine and twelve so looking at the fact we've got two ones it is not a function i have repeated input okay part d i have 3 5 4 5 5 10 8 3 yes it is a function and i have unique input part e and f equations so what i do when i look at an equation or i'm thinking about an equation whether or not it's a function i think of the picture and what the function represents so if you remember from grade 10 x squared plus y squared equals 36 is a circle actually with a radius of 6 so a radius of square root 36 is this a function no it's a circle and that is reason enough for you to tell me it's not a function what type of graph is this well i hope that you're sort of saying yourself it must be a quadratic all quadratics are functions so i can say yes it's a function because it's a quadratic part g okay i've got my vertical line as i move from left to right many times i hit two x's at one time no and it fails the vl t how about this guy let me steal this vertical line over here from left to right those uh that should be a smooth curve left to right a quadratic it is a function and it passes the vlt i could also just say it's a quadratic there then we get into function notation so this this will feel very strange to begin with but in a couple weeks you'll be very very used to it and and quite comfortable so we're moving away from using y so we're taking what we're used to in y equals mx plus b form or y equals form to represent equations and relations and we're moving into this function notation it's a naming system and how it works okay so f bracket x reads a couple different ways f of x f at x are the most common f of x f at x and it basically gives us the ability to evaluate and name functions so here are the details x is your input f at x is the output of the function it does not mean f times y or sorry f times x it is basically giving your y value okay which is your output and f is the name of your function and we use most common names are f g h um we can use them in problem solving we can change the x to a t it becomes very very helpful for using different functions and naming them let's do a few examples i've been asked to evaluate f at zero which means if i'm writing f at zero i'm taking my function and i'm replacing okay i'm replacing my input with the value in my bracket which is a zero so i leave the negative 3 i replace the x with a 0 and then i plus 6 where i add 6. negative 3 times 0 is 0 and i add 6. so f at 0 equals 6. f at negative 4 means that i take the negative 4 and i sub it in where the x is as the input value 12 plus 6 equals 18. therefore f at negative 4 equals 18. f at a plus one follow the rules don't be thrown off because it's an a plus one i'm going to okay i'm going to take my a plus one and i'm going to sub it into where the value is for x so a minus 1 and plus 6. so negative 3a plus 3 plus six equals negative three a plus nine a couple more down here f of two minus f at one you have a couple options on what to do in this case what i want you to recognize you cannot bring these two inside and evaluate f at one okay they do not mean the same thing f 2 minus f at 1 means take the answer from f at 2 so plug 2 into your input and subtract i'm going to use brackets here so i don't make any mistakes f at 1 is sub 1 into the input and then i have negative 6 plus 6 which is 0. and i have negative three plus six which is three and i have as my final answer negative three three times f of okay i leave the three alone f at five is negative three bracket five plus six okay now evaluate i probably would clean up the inside first and evaluate so that i can sort of keep track so negative 15 plus six is negative nine now i times by three negative twenty-seven be careful now because you're done here i'm not dividing both sides by three and trying to solve for f at five i've been asked to evaluate three times f at five second example so i'm given a quadratic and i'm asked to determine the following the rules are exactly the same whatever is in the brackets is what i will replace my variable so g at two equals negative two times two squared plus two times two minus six so negative two times four is negative eight plus four minus six gives me negative uh what's that negative four minus six negative ten okay g at 2 plus g at negative 1. now i've already done g at 2. so i can make use of that and i can say all right g at 2 is negative 10 and i'm going to add i'm going to use brackets just to be careful i'm going to sub negative 1 and now and simplify this is squared so it's a negative one times negative two i get negative two minus six negative 10 again and my g at 2 plus g at negative 1 is negative oops negative 20. okay i'm running out of room for part c here there we go we have a little bit more room so now i'm going to sub in so my input is a plus 5 so i have negative 2 a plus 5 squared plus 2 times a plus 5 minus 6. negative 2 careful expanding a plus 5 squared b squared right that is the same just as a reminder it is the same as a plus five times a plus five somebody along the way may have told you a little shortcut for squaring a bracket i can square the first thing i can square the last thing multiply them together and double it to get the middle so i've got a 10a in the middle that's just a little expanding trick 2a plus 10 minus 6. negative 2a squared minus 28 minus 50. plus 2a plus 4 right 10 take away 6 is 4. so i've got no other i don't have any other a squareds negative 28 plus 2 so negative 18 a and negative 50 plus 4 negative 40 6. okay next on the agenda is to take a look at questions that are slightly different in their appearance whoops and it's a question where i have been given this table and asked to solve some to evaluate using these function or this function notation so the question has said to me evaluate or has asked me evaluate h at 20. so this guy is my input so i go to my x it's input 20 oh my output is zero h at negative one go look at my x's i don't have negative one that means that this must not exist okay or undefined would also work h at eight i go down to eight and i see four so h at eight equals four part d find the values where h at x equals nine so now i'm looking for x's so that my h at that particular x equals nine so these guys have to be the same and my output is 9. so now i'm looking in this column and i see that when x is 2 my y is 9 therefore x is equal to 2. there could be more than one answer for a question like this because the y's can be repeated last question so this is a bit of a thinker so it's asking me the values of x such that f at x equals negative 20. so f at x equals negative 20. that means that this guy is negative 20. i'm hoping this looks really familiar it might take a minute to sort of sink in and and for you to recognize it's a quadratic and you have all sorts of tools from last year in grade 10 that help you to solve a quadratic and the game always was get a zero on one side that's the first thing i never want to work with with values on both sides of the equal sign so minus fourteen plus twenty i have zero equals x squared plus five x plus six hopefully you're saying oh now we just factor that you got it x plus 3 and x plus 2. at this point last year i would have said to you branch off take each of your brackets and set them equal to zero so one solution would be when x is negative three or second solution when x is equal to negative two okay x equals negative three and x equals negative 2. i do believe that oh we do have one more a graph to look at i'm not sure if this is on your paper or not but we will take a look at it both questions are asking me to figure out f at one both graphs rather so can we figure out our graph here this guy must be zero zero just to give us an idea this one's a little bit easier zero zero so the question is asking for f at one so i go over to one and i want to go and find my function which it looks as though this guy is equal to two if at 1 i go over to 1 and it looks as though my function's value or the second graph is negative 2 and that finishes off the lesson give the homework a try