### Stony Brook MAT 123 Fall 2015 Inverse Functions

 Start in this module we're going to learn how to find the inverse function so if you watch the composition of functions we saw that the inverse function you can you can tell us two functions are inverses if f of g of x is the same as G of f of X and they're both equal to X or in other words F of G of x equals x g of f of x equals x then f of X and G of X are inverses inverse function so what 0:41 does that mean to be an inverse function inverse function kind of undoes the operation of the other function so square root of x and x squared are inverses cube root of x cubed is actually slightly better example so if f of X is X cubed, f of 2 is 2 cubed is 8 and then g of x the cube root of x and g 8 is the cubed root 1:13 eight which is two notice first you plug in two and you get 8 and then you plug in 8 and get back to two so g of x and f of x sort of undo each other's operation that's one way to think about inverse now y is x squared squared of X not going to work out so well...well 1:32 let's see let's say that f of x is x squared g of x the square root of x if i plug in 2 x squared 4 and g4 is the square root 2, square four which brings me back to two but what if i do F of negative 2, well f of negative 2 is negative 2 squared which 2:02 is also four but G of negative 4, 4 sorry is the square root for which is to not negative 2 so you have to be careful when you have functions and their inverses what happens is there are functions that are their numbers that are the domain of f of X that will not be in the range of g of x so functions and their inverses the domain of the one function is the range of the other and 2:34 the other way around so the input to the first function because the output of the second function and vice versa another way to think about inverses is if f of X goes through the point a comma B and the F inverse of X which we write this 3:00 negative 1 notation will go through the point B comma A so notice you put a you get out b here if you plug in b get back to a practically that has the effect of you take the line y equals x and they are reflections over the line y equals x so if you have the point a comma B will be here and the point b common a be about there if you 3:30 do a straight line will be perpendicular to y equals x distance will equal this distance a little mathy but that's what's going on with an inverse so the reflections over the line so one function looks like this other function would look like that you can imagine if you fold it on the line that they go on top of each other ok so how do we find the inverse function well it's not always easy to do but if you have a simple function say you have oh you want to find the inverse function of that 4:09 that this simple way to do is with an algorithm first replace f of X with y so replace okay you have two steps you switch x and y and then you isolate y so here 4:45 if you want to switch x and y so this would have been a written as to 2X plus 3 great so step one switch X&Y so it becomes X is 2 y plus 3 and now you want isolate y so subtract 3 from both sides and divide by two 5:07 that's y and thats the inverse function so the inverse of f of x equals 2x plus 3 is F inverse of X is X minus 3 over 2 let's verify that so f love F inverse of X would be you take X minus 3 over 2 and 5:38 plug it in there two times X minus 3 over 2 plus three and you do the cancelations of x and you get x and then if you do f inverse of f of X now you'll take FX and put it in there so you'll get 2x plus 3 minus 3 over 2 which is x so remember think that works inverses of 6:05 each other so let's do it again with a slightly messier function so let's say f of x is 5x cubed plus 4 over 7 that looks very bad but it's not so 6:32 replace f of X with Y so Y is 5x cubed plus 4 over 7 our first steps which x and y x is 5 y cubed plus 4/7 and now we're going to isolate y so cross multiply subtract 4 from both sides / 5 and take the cubed route and that is the inverse function 7:18 remember what I said the one function undoes the others if you think about it this function you plug in the number and what you doing first you cube it then you times 5 then you 7:32 add 4 the last you divide by 7 so now here you do the other working divided by seven so the first thing you do is you multiply by 7 then you add 4 i subtract 4 nd multiplied by 5 and now you divide by 5 and then the last thing you did was cubed it and last thing you do here is cubed root it and that's another way to think about inverses so we're do inversus show up, so where i showed you square root cube root fourth root those are inverses of squared cubed fourth sine cosine and tangent all the 8:04 trig functions have inverses inverse sine inverse cosine inverse tan and logarithms and Exponential's are inverse are the main places you're going to see inverse functions in precalculus and calculus