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Current time:0:00Total duration:4:12

Verifying inverse functions by composition: not inverse

Video transcript

let's say that f of X is equal to two X minus three and G of X G of X is equal to one-half X plus three what I want to do in this video is evaluate what f of G of X is f of G of X and then I want to evaluate what G of f of X is so first I want to evaluate F of G of X and then I'm going to evaluate the other way around I'm going to evaluate G of f of X but let's not get F of G of X purse and I like always encourage you to pause the video and see if you can work through it so this is going to be equal to F of G of X is going to be equal to well wherever we see the X in our definition for f of X the input now is G of X so we'd replace it with the G of X it's going to be 2 times G of X 2 times G of X minus 3 minus 3 and this is going to be equal to 2 times well G of X is all of that business 2 times 1/2 X plus 3 and then we have the minus 3 minus 3 and now we can distribute this to 2 times 1/2 X is just going to be equal to X 2 times 3 is going to be 6 so X plus 6 minus 3 and so this is going to be this is going to equal X plus 3 X plus 3 all right interesting that's F of G of X now let's talk about what G of f of X is going to be so G of our input instead of being instead of calling our input X we're going to call our input F of X so G of f of X is going to be equal to is going to be equal to 1/2 times our input which in this case is f of X 1/2 times f of X plus 3 plus 3 you can view the X up here is the place holder for whatever our input happens to be and now our input is going to be f of X and so this is going to be equal to 1/2 times what is f of X it is 2 X minus 3 so 2 times X minus 3 and then we have a plus 3 and now we can distribute the 1/2 1/2 times 2 X is going to be X 1/2 times negative 3 is negative 3 halves and then we have a plus 3 so let's see 3 is the same thing as 6 halves so 6 halves minus 3 halves is going to be 3 halves so this is going to be equal to X plus 3 halves so notice we definitely got different things for f of G of X and G of f of X and we also didn't do a roundtrip we didn't go back to X so we know that these are not inverses of each other in fact we just have to do either this or that to know that they're not inverses of each other so these are not inverses so let me write it this way f of X f of X does not equal does not equal the inverse of G of X does not equal the inverse of G of X and G of X does not equal the inverse of f of X does not equal the inverse of f of X in order for them to be inverses if you have an x value right over here and if you apply G to it if you input it into G and that that takes you to G of X so that takes you to G of X right over here so that's the function G and then you apply F to it you would have to get back to the same place so G inverse G inverse would get us back to the same place and clearly we did not get back to the same place we didn't get back to X we got back to X plus 3 same thing over here we see that we did not get by we did not go get back to X we got back to or we got to X plus 3 halves so they're definitely not inverses of each other