Proving that two functions are inverses algebraically. So if we want to prove that two functions are inverses what we have to do is a composition of the two of them in both orders and see that we get x out.
So for this example I have two different problems up here and I want to prove that they are inverses. So the first thing I want to look at is f of g of x which is telling me I’m taking g and I’m plugging it into f. So wherever I see an x and f in goes g. Oops that’s a minus, no that’s a plus I was right and then minus 3. So this one simplifies quite easily, multiplying by 6 dividing by 6, these are going to cancel, x plus 3 minus 3, our 3’s cancel just leaving me with x.
So I did one step, I did the composition of f and g I also need to do the opposite to make sure that the same thing x just comes out. So for this one g of f of x, now we are taking f and plugging it into g we end with, let’s try that again we are taking f plugging it in over there so we end up with 6x minus 3 plus 3 over 6 so there is our f function. So we have minus 3 plus 3, 3’s cancel 6 x over 6, 6’s cancel just leaving us with x.
So composition of f and g both orders x came out so we have in fact proved that they are inverses. So always make sure you do both orders because it is possible for one to come out x the other one not to which will tell us they are not an inverse.
So composition you know we in fact should go back and refresh how to do these compositions everything else should be pretty straight forward.