### Stony Brook MAT 123 Fall 2015 Composition of Functions

 Start this module will be how to find the composition of functions what does that mean, the composition of functions? that means that one function is inside another function so you have two functions f of x and g of x and the composition means say you take g of x and youre gonna piut it inside f of x thats means, the reality is you take some value of x 0:32 first you do g to it you take that answer and then you preform f so for example if f of x is x squared plus 4 and g of x is x-3 and you want to find the composition f of g of f of g of 5 1:02 so those are the notations f of 5 goes into g of 5 that can also be written as f circle g of 5 the circle means do the composition you start from the right side and you do it to the left sides what if f of g of 5, well first we would need to find g of 5 cause 5-3 equals 2 1:30 then we do f of 2 because g of 5 is 2 and thats 2 squared plus 4 equals 8 therefore f of g of 5 equals 8 so again whats happening is you are taking the function and first youre preforming g and then youre preforming f you take the output of g and that becomes the input of f 2:01 so lets do another one lets say f of x is the square rot of x+3 and g of x is x squared minus 1 and f of g of 2 would be what? first we find g of 2 which is 2 squared minus 1 2:30 which is 3 then we find f of 3 which is the square root of 3+3 which is the square root of 6 now we can have a little problem here suppose you wanted to find f of gof 0 g of 0 0 squared minus 1 which is -1 and we cant find, oh we could never mind 3:00 f of negative 1 is the square root of -1+3 is 2, we would have to be careful that g of x doesnt give us a number that makes f of x not work, that doesnt fit into the domain of f of x so in this case its not a problem suppose we wanted to find f of g of 2 we wanted to find g of g of 2 okay so first we find of of 2 3:31 which is the square root of 2+# which is the square root of 5 and then we would find g of the square root of 5 which is the square root of 5 squared minus 1 which is 4 notice we can do f of g of 2 or g of f of 2, we dont get the same answer what else should we do? well what if instead of giving you numbers we want find f of g of x 4:02 so f of g of x means we are going to take g of x and replace it inside of f so if f of x was the square root of x plus 3 and g of x is x squared minus 1 then f of g of x would say take x square minus 1 and plug it into x so you would get the square root of x square minus 1 4:30 plus 3 which is the square root of x plus 2 and we can find f of g of x we go the other way, now we are gonna take the square root of x plus 3 and plus it in to x so that would be square root of x plus 3 squared minus 1 which is x plus 3 minus 1, which is x plus 5:01 2 what else can we do a composition function well what if theres 3 functions suppose we have f of x is 4x+! g of x is 1/x and h of x 5:31 is 4-x squared and i wanted to find f of g of h of x lots of ofs so first im gonna have to find h of x first h of x 6:01 sorry lets say f of go of h of 6 so first h of 6 is 4-6 squared which is 4-36 which is -32 then i want to find g of -32 6:30 which is 1/-32 and finally i want find f of -1/32 which is 4 times -1/32 plus 1 which is 7/8 you can do f of g of h you can do you can do h of f you can do all sorts of things, of course you can do 7:00 f of g of h of x, now lets do that so first we tae h of x and put it inside g of x so g of h of x is 1/4 minus x squared so we took x and we plugged in 4-x squared and then we are gonna find 7:31 f of 1/4-x squared because thats f of g of h of x and that is 4 times 1/4 minus x squared plus 1, you can combined that into one fraction but were not gonna bother notice how earlier we said f of g of x and g of f of x is not necessary the same thing suppose 8:06 f of x is 4x plus 3 and g of x is x minus 3 over 4 and f of g of x is 4 times x-3/4 plus 3 cause we took g of x 8:30 and we plugged it into x so that, the 4s cancel and you get and you get x-3+3 which is x what if i did g of f of x well now im gonna have 4x plus 3 goes in for x minus 3/4 that gives you 4x/4 which is x so if f of g of x 9:00 gives you x, and f of g of x also gives you x then theses functions are inverses of eachother and thats gonna be the next module