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Current time:0:00Total duration:6:00

Restricting domains of functions to make them invertible

Video transcript

- which intervals could we restrict f of X is equal to cosine of X minus PI over 4 so that f of x is invertible and they show us what cosine of X minus PI over 4 what it looks like the graph of it so let's just think about what it means for a function to be invertible so a function is a mapping from a set of elements that we would call the domain so let me my pen is off today so let's see if it works okay so this right over here is our domain and this over here is our range our range and a function maps from an element in our domain to an element in our range that's what a function does now the inverse of the function maps from that element in the range to the element in the domain so that over there would be f inverse so that's the direction of the function that's the direction of F inverse now a function is not invertible one of the situations in which a function is not invertible you could have a function where two elements of the domain map to the same element of the range so these both of these elements map to that element of the range but then so both of these are the function but then if this is the case it's going to be you're not going to be able to create a function that maps the other way because if you input this into the inverse function where do you go do you go to that element of the domain or do you go to that element right over there and so one way to think about it is you need a one-to-one mapping for each element of each element of the domain range there's only one element of the domain that gets you there or another way to think about it you could try to draw a horizontal line on the graph of the function and see if it passes if it crosses through the function more than once and you could see that this is indeed the case for this function right over here if I did a horizontal line right over here now why is this the issue well this is showing us actually let me show doing a number that's a little bit easier to look at so let's say I drew the horizontal line right over here now why is this horizontal line an issue well it's showing us that just even what the part the part of the domain that's being graphed here that there's several points that map to the same element of the range they're mapping 0.5 0.5 if you this value right over here when you take when you input the F of that is equal to 0.5 F of this right over here is equal to 0.5 F of this right over here is 0.5 so if you have that if you have multiple elements of your domain mapping to the same element of the range so the function will not be invertible for that domain so really what we're going to do is we're going to try to restrict the domain so that for that domain if I were to essentially apply this what I'd call the horizontal line test I'd only intersect the function once so let's look at the or the graph of the function once so let's look at these choices so the first one is an open set from negative 5 PI over 4 negative 5 PI over 4 so that's PI that's negative PI and another fourth of Pi so that's I think starting right over here going all the way to negative 1/4 PI so that's this this domain right over here let me do this in a new color so that's this and this does not include this does not include the two endpoints so here I can still apply the horizontal line and in that domain I can show that there's two there's two members of the domain that are mapping to the same to the same element in the range and so if I'm trying to construct the inverse of that what would that what would this element is I guess it's negative 0.6 what would that f inverse of negative 0.6 be would it be this value here or would it be this value here so I would rule this one out so let's see negative PI to PI so negative PI to PI I'll do this in I'll do this in this color right over here negative PI to PI this is a fairly so once again over this is a close so we're including the two boundaries we're including negative PI and PI in the domain but once again over that interval I could apply my horizontal line here and notice or actually I could even apply the original one that I did that I did in blue and notice there's multiple elements in the domain that map to say zero point five so what would F inverse of 0.5 be you can't construct a function where it Maps only to one element of the domain so we could rule this one out right as well now negative 1/2 PI to positive 1/2 pi so negative 1/2 pi so that is so let me so I'm running out of colors so negative 1/2 pi to positive 1/2 pi this one is interesting if I apply horizontal line there they're there so let's see but if I apply a horizontal line right over here I do intersect the function twice so I have two members of this domain mapping to the same element of the range so I could rule that one out as well and I'm not left with one last choice so I'm hoping this one will work out so 1/2 PI then it's an open set so 1/2 PI right over there 2 5 PI over 4 5 PI over 4 so that's PI and another one for it so that's right over there and let's see this is if I were to look at the graph here it seems like it would pass the horizontal line test at any point here I can make a horizontal line over that domain actually let me do it for the whole domain so you see that for the whole domain and I'm only intersecting the function once so for every every element of every element of the range that we're mapping to there's only one element in our domain that is mapping to it it's passing our horizontal line test so I would check it this one right over there