Inputs & outputs of inverse functions

you may by now be familiar with the tape the notion of evaluating a function at a particular value so for example if this table is our function definition if someone were to say well what is f of negative 9 you'd say okay if we input negative nine into our function if X is negative nine this table tells us that f of X is going to be equal to five and you might already have experience with doing composite functions where you say f of let's say actually F of F of negative nine plus one so this is interesting it seems very daunting but you say oh well we know what F of negative nine is this is going to be five so it's going to be F of five plus one so this is going to be equal to F of six and if we look at our table F of six is equal to negative seven so all of that is a review so far but what I want to now do is start evaluating the inverse of functions and this function f is invertible because it's a one-to-one mapping between the X's and the F of X's no two X's map to the same f of X and so this is an invertible function so with that in mind let's see if we can evaluate something like f inverse f inverse of 8 what is that going to be and I encourage you to pause the video and try to think about it alright so f of X just as a reminder of what functions do f of X is going to map from this domain from a value in its domain to a corresponding value in the range in the range so this is what F does so this is domain domain and this right over here is going to be the range now f inverse if you pass it the the value in the range it'll map it back to the corresponding value in the corresponding value in the domain but how do we think about it like this well f inverse of 8 this is whatever maps to 8 so if this was 8 we have to say well what mapped to 8 what we see here f of 9 is 8 f of 9 is eight so f inverse of 8 is going to be is going to be equal to and actually we do this in that same color is going to be equal to nine and if it makes it easier we could actually construct a table here and this is actually what I probably would do just to make sure I'm not doing something strange where I could say X and F inverse of X and what it essentially do is I'd swap these two columns so f of X goes from negative 9 to 5 f inverse of X is going to go from 5 to negative 9 all I did is I swapped these two now we're mapping from this to that so f inverse of X is going to map from 7 to negative 7 notice instead of going from this mapping from this thing - that thing we're not going to map from that thing to that thing so f inverse is going to map from 13 to 5 13 to 5 is going to map from negative 7 to 6 negative 7 to 6 it's going to map from 8 to 9 8 to 9 and it's going to map from 12 to 11 from 12 to 11 so let's see did I do that right it looks like I got all of yep so all I did is really I just swapped these columns it's mapping the F inverse maps from this column to that column so I just swapped them out so now it becomes a little bit clearer that way and you see it right over your F inverse of 8 if if you input 8 into F inverse you get 9 so now we can use that to start doing fancier things we can evaluate something like F inverse F inverse of actually list let's evaluate this let's evaluate F of F of F inverse of let's see F of F inverse of 7 what is this going to be well it's first evaluate F inverse of 7 F inverse of 7 maps from 7 to negative 7 maps from 7 to negative 7 so this is going to be F of this stuff in here because we do in yellow this stuff in here F inverse of 7 we see is negative 7 so it's going to be F of negative 7 and then to evaluate the function well F of negative 7 that's just going to be 7 again and that makes complete sense we essentially went we mapped from 7 F inverse of 7 went from 7 to negative 7 and then evaluating the function of that went back to went back to 7 so let's do let's do one more of these just to really feel comfortable with mapping back and forth between these two sets between applying the function and the inverse of the function so let's evaluate what would in purple F naught that was purple so it's going let's try to evaluate F F inverse of F inverse of 13 of F inverse of 13 what is that going to be I encourage you to pause the video and try to figure it out well what's F inverse of 13 well that's looking at this table right here F inverse goes from 13 to 5 and you see it over here F went from 5 to 13 so f inverse is going to go from 13 to 5 so this right over here F inverse of 13 is just going to be 5 so this whole thing is the same thing as f inverse of 5 and F inverse of 5 well F inverse goes from 5 to negative 9 so this is going to be equal to negative 9 once again F inverse goes from 5 F goes from negative 9 to 5 so f inverse is going to go to from 5 to negative 9 so at first when you start doing these kind of inverse and function and inverse of a function it looks a little confusing hey I'm going back and forth we just have to remember a function maps from one set of numbers to another set of numbers and the inverse of that function goes the other way so if we if the function goes from 9 to 8 the inverse is going to go from 8 to 9 so one way to think about is you just kind of switch these columns hopefully that that clarifies more things that it confuses