### Stony Brook MAT 123 Fall 2015 Lecture 10b: Review for first midterm. September 26, 2016

 Start lets do some review stuff so whats going to be on the exam? its gonna be some f of x stuff, you know f of x g of x putting together taking the apart theres gonna be domain and, no range just domain finding inverse of functions ill write these down, okay your gonna need to know f of x stuff that is domain 0:31 inverse composition composition is f of g and g of f, that kind of stuff your going to have umm trigonometry your gonna need to know special angles 1:01 the 6 trig functions and your gonna have some word problems i think thats about it composition is when you do f of g of x or g of f of x oh your gonna have to do some of quations stuff here, gonna have to do f of x plus h 1:33 minus f of x trigonometry lets see we did the 6 functions, soh cah toa we did the word problem i think thats pretty much whats on the exam i cant really think of anything else we can do it theres not gonna be transformations on the test i know that makes you sad i can add them now at the last minute, no? 2:05 so lets practice some stuff theres 172 questions on the exam each of them have 6 parts some of them have sub parts i forget its like 9 its about 4, dont hold me the number cause i dont have them with me theres about 4 for the minimum confidence part and then theres like four or 5 for the advance part so remember your first goal is to pass minimum confidence 2:32 you guys should be able to do that just fine once you did that you lock in your minimum stuff and now you go to the advance stuff shouldnt be to bad not a lot of meterial on the test i didnt want to make it to hard sorta ad, do i actually feel bad? i dont feel bad i can go to sleep at night knowing youre crying 3:05 suppose you would have f of x equals and g of x ugh why dont you find 3:44 alright heres three things to do that should keep ypu busy for a couple minutes so we ask for g of f of 2 what do you have to do? 4:00 so you have g of x of 2 first you have to find f of 2 f of 2 equals 5 times 2 cubed minus 4 times 2 plus 1 i think thats 33 then you fin g of 33 which is 2 4:31 over 33 minus 6 which is 2/27 thats fine thats 27 you get that one? good f of g of h now we go the other way first we find g of 8 then it equals 2/8-6 i think that equals 1 5:06 and then f of 1 is 5 times 1 cubed minus 4 times 1 plus 1 equals 2 everybody happy so far if i ask for f of g of x which i didnt ask for 5:32 now what would i do, if we do f of g of x you take g of x and put it inside f of x that would be 5 times 2/x-6 cubed minus 4 times 2/x-6 plus 1 and you dont have to simplfy that thats very hard to simplfy yes you want to simplify 6:08 i can show you g of g how bout i show you that oin friday gonna throw one to if you want to find f of f or g of g you literally take f and put that in where ever theres x so if you were doing f of f of x 6:30 everytime you have an x here you ave to put in 5x cubed minus 4x plus 1 and here if you have g of g, where x you put 2/x but im not concerned about those alright if you want to do g of f of x now you do it the other way, where ever theres an x you put in f so thats 5x cubed minus 4x minus 1 minuss 6 7:01 so far so good? now i just realized the inverse is really hard to find wow alright ignore that and for those of you watching this at home on the video were not gonna do that problem were gonna do a different one its hard to solve because its a cubic i think that if you think you go that answer yu didnt im not gonna insult anybody but youd have to isolate 7:32 y and the cubic which means youll have to solve the cubic which is quite messy so i wasnt concentrating on that alright so lets give you a simpliafr one okay lets say f of x 5x cubed minus 4 now find the inverse 8:01 and what are you gonna do youre gonna switch x and y and isolate y alright im gonna let y equal 5x cubed minus 4 switch x and y and isolate y how would we isolate y? well first wed add 4 to both sides 8:30 and we would divide by 5 x+4 over 5 equals y cubed then you would take the cubed root in fact f inverse of x, and thats all youd have to write so on your exam theres going to be a box for every question where you put in the answer so the tas are gonna grade 100s of your exams, each ta gets a question 9:01 so you get consistancey, so the tas are gonna get tired theyre not gonna fight you way through all your scribbling so make sure you put your answer in the box so they can find your answer cause many of you come back to us and say see over here theres my answer its not neccasarily going to fly thats all your gonna have to write nothing more complicated then that so we got the idea of these 9:32 lets do a domain problem 10:20 alright so theres 4 different f of x's find the domain 10:31 alright so four different types of things where were going to find the domain but they have the square root notice its the square root not the cubed root you can take the cubed root of anything you want, you can take the odd root of any positive negative or zero right all real numbers so if thats the cubed root of 8x-2 would be 0 but it does if it the square root of 2 well all you do is 8x-2 11:01 has to be greater then or equal to 0 well is 8x-2 is greater then 0 the 8x has to be greater then or equal to 2 x has to be greater then or equal to 2/8 or x has to be greater then 1/4 now im sure your gonna ask me do i have to put this in interval notation no as said before but just in case you dont have to do interval notation you should know interval notation 11:31 and you should love interval notation well you dont have to love it you just have to know it but you dont have to do it on the exam and if your tas says you do welll you dont every once in a while i get oh may ta said to do, well im in charge i get to decide 12:00 f of x is 1/5 plus 1 what can you not do with 1/5xplus1 you cant have 0 in the denominator so any ration expression the denominator cannot be 0 the numerator can be 0 but in this case its always 1 so 5x plus 1 cannot equal 0 so x cannot equal -1/5 how we doing so far? 2 for 2 12:30 now for the entertaining ones, what if i combine theses so if you have f of x is square root of 8x plus 2 over 5x+1 now you have 2 problems the square root that still cant be negative you still cant get the square root of negative so 8x plus 2 still has to equal 0 so x has to be greater then or equal to a quarter right 13:02 that should be minus 2 how did that magically become plus two oh well plus 2 so thats minus a quarter and also x cannot equal -1/5 so you need both of those you technically should write the word and why did i get ride of the radical here? 13:30 well remember you cant take the square root of a negative numbe so to find the domain you just take whats under the radical and that has to be greater then or equal to 0, just trying to figure out what to plug in what youre allowed to take the square root of so youre not allowed to take the square root of negative numbers so you take this has to be greater then or equal to 0 14:02 so both of these have to be true now what about the other way around what if i had f of x is 5x plus 1 ovr square root of 8x plus 2 now he square root of 8x plus 2 again you cant take the square root of a negative number this also cant be 0 now because then you have 0 in the denominator 14:30 so 8x plus 2 has to be greater then 0 x has to be greater then -4 so it cant equal -1/4 because if it equal a negative 1/4 youd have 0 in the denominator and you cant have 0 in the denominator should i repeat that? when you see the square root you cant take the square root of a negative number so just as before 8 x plus 2 is greater then or equal to 0 15:00 and thats greater then or equal to -1/4 however when x equals negative 1/4 you now have 0 in the denominator and you cant have 0 in the denominator so you have to throw that one number out and you just have greater then -1/4 so thats the different from having the radical in the numerator and radical in the denominator in the numerator you can have 0 but in the denominator you cant have 00 so you only have greater then sign 15:31 so range changes just a little bit now we have 0 in the numerator you can have 0 in the numerator so this you can ignore make sure you understand the differnece amoung the four of these lets do something else 16:24 suppose f of x is 6x squared plus 4 16:30 suppose f of x squared plus 4 find f of x plus h minus f of x 17:16 alright so if this is f of x and f of x plus is 6 x plus h squared plus 4 now we wnat the difference between the 2 17:36 so you go this one minus multiply that out you get 6 times x squared plus 2xh plus h squared plus 4 minus 6x squared 18:02 minus 4 so thats what you get when you foil out the x plus h now distribute you get 6x suqared plus 12x h plus 6/h squared plus 4 minus 6x squared minus 4 the 6 x squared cancel and the 4s cancel because i said to you guys 18:34 the only thing left the only terms left the only that contain at least 1 h so these cancel and these cancel so you get 12x h plus 6h squared lets try one more make a messier one see i like to do an easy one and a hrad one 19:17 find f of x of that one so f of x plus h 19:30 4 x plus h cubed minus 3x plus h plus 5 so were gonna take this one and subtract this one so 4 xplus h cubed is x cued minus 3xh plus 3xh squared plus h cubed minus 3x minus 3h plus 5 minus 4x suqraed minus 3x plus 5 20:03 that is 4x cubed plus 12 x squared h 12 x h squared plus 4h minus 3x minus 3h that was cubed that would be 4x cubed plus 3x minus 5 cancle the 5 the 3x 20:32 and youre left with 12x squared h plus 12 x h squared plus 4h cubed you do not have to factor that alright everyione get that? okay wednesday we will do trig stuff