Function invertibility

Voiceover:We've seen that a function maps elements from its domain. If you give it an element in its domain it will then map it to an element in its range. So for example, if this is the domain. Let me do this in this orange color. If this is the domain of a function and let me just pick out some random elements right over here. So let's say that this is one. This is negative one. This is seven. And this is pi. Let's say these are all members of the domain of f. And let's say that f and let's say that this right over here is the range. So this is right over here is the range. And so let's say that f of one, f of one. So if you input one into the function, the function maps one to the number negative three. And let's say it maps the number negative one So f of negative one is equal to, is equal to seven. And let's say that f of seven, f of seven is equal to 10. But let's says that f of pi is also equal to 10. So this is the function map. Again this is a legitimate function. Remember in order to be a function for any input it has to map it to a unique it needs to map it to only one output. This wouldn't be a function if it did something like this. If you input pi instead of saying you're gonna get 10 instead of saying that f of pi is equal to 10. This is clearly a function. You're mapping from pi to 10. For the input pi you know you're gonna get the output 10. But if you were doing some type of a relationship where you said oh well if you put in a pi sometimes it's a 10 and sometimes it's an 11. Now this would not be a function cause you're mapping pi to two different values right over here. So that's not allowed if you're thinking about a function. For any input you have to map it to only one, only one output. Now you could have two inputs that map to the same output because this is predictable. You know that f of seven is 10 and you know that f of pi is 10. Well what I wanna do is now think about when does a function not when it is not invertible? We've already seen that the inverse of a function should map between an element of its range and the corresponding element in the domain. So the inverse of f should map three, negative three to one. So f inverse. So because because f of one, f of one. Actually let me write it this way. Actually I'll just write it. Because f of one was equal to negative three f inverse, f inverse of negative three. And the inverse function is also a function. So it has to have the same properties of a function. It should map to only one thing. F inverse of negative three is going to be equal to one. F inverse of seven is going to be equal to negative one. F inverse, this is f inverse right over here. But now we have something strange happening if we think about let me write this down. F inverse of seven is equal to negative one. That makes sense because we already knew that f of f of negative one is equal to seven. But now something interesting happens over here. What would f inverse of 10 be? Well f inverse of 10 if I'm trying to map it 10 there's two elements of the domain that when I applied the function got us to 10 in the range. So if you're taking, if you're trying to see whether f is invertible then the inverse function should map this element to only one element of the domain. But its mapping it to two elements of the domain. So is this equal to seven or is this equal to pi? And because the inverse mapping can't be done where you only map it with a unique thing this function is not invertible. This is not not invertible. The inverse mapping, the inverse mapping from the range to the domain the inverse relationship is not a function. It does not it can't take an element in our range and map it to a unique element to the domain. For 10 we don't know. Does that map to pi or does that map to seven? So this function f is a function. For any member of its domain it goes to a unique it goes to its a very well defined element in the range. But the other way around isn't true. For the inverse function for any element in this set it doesn't go to only one element in that set. This 10 could be mapped to either pi or seven so f inverse so another way to think about this is not an invertible f. F is not invertible. F is a function but the inverse mapping is not a function so we do not so we would say that f is not invertible.