 So when studying inverse functions a very important property about inverse functions is the so-called inverse function property Which you see written on the screen right now the inverse function property What the inverse function property tells you is the following if we have a function F And we compose it with its inverse function So F compose with F inverse will just spit back out an X where X was the number we started with and if you go the other Way around if you do F inverse first and then F F inverse compose with F If you apply that to the number X, we'll just give you back the number X now notice the function that assigns X to itself This is what we call the identity function and just identifies the number you gave it So if you give it one it says one if you give it two it says two if you give it Pikachu It says Pikachu right it just identifies what it sees there when you compose a function with its inverse function You get back just the identity you get back the original number And the idea is kind of like the following if you have a machine, right? Let's take our let's take our soda machine that we talked about in a previous video, right? If we have a machine where you put inside of an input you put inside of it an empty bottle It fills it up with soda look look look look look look look it then spits out It then spits out a filled bottle of soda ready to go to the store, right? We as the consumer do the inverse operation the the factory fills the bottle with soda And then we as the consumer drink the soda and thus sending it back to An empty bottle, right? So the F inverse is going to reverse this process, right? So instead of things going into on the right F inverse is going to send things in from the left and the output on the right So when you do these things together in tandem, it's as if nothing happened, right? It'd be like saying oh, what's x plus three? Minus three well whatever the number x is it doesn't matter if you add three then subtract three They're gonna cancel each other out and give you just back an x This is what inverse functions are about when you do them together It's as if nothing happened the net change is nothing And so let's do a more complicated example here Let's take the function f of x equals 2x plus 3 and here's another function g of x equals 1 half x minus 3 halves I claim that these two functions are inverse functions of each other in order to show that two functions are inverses What you're gonna do is you're gonna compose them together So compute f composed with g of x here, right? So this means you put g of x inside of f of x well f of x just means 2x plus 3 So f of g would be 2g of x plus 3 and then inserting the definition of g of x which we see right here We're going to get 2 times 1 half x minus 3 halves and then we add 3 to that now let's work to simplify this expression To begin with we can distribute the 2 Which is pretty nice because all the 1 halves that are in play there You're gonna get 2 times 1 half which is just one so you just get an x You're gonna get 2 times negative 3 halves, which is just negative 3 and then you get a plus 3 And what do we say a moment ago negative 3 plus 3 they cancel out and you just get back at x When you do f compose with g it's as if nothing happened to the number It just one on a big big circle On the other hand we have we want to check both directions here f can g compose with f of x That is we're gonna put f inside of g g of f Which conversely if you want to that just means you're gonna take g of 2x plus 3 you could evaluate at first and put that inside of the formula for g you get 1 half times 2x plus 3 minus 3 halves and So simplify and I'm going to distribute the 1 half in this situation Like so that would give us a well 1 half times 2 like we saw before is just a once You just get an x you get 1 half times 3 which is 3 halves and then you're going to subtract 3 halves Which those are going to cancel out and you're left again with just an x and so you can see in this situation that No matter which order you compose them f of g or g of f in both situations You get x just back so the number x is unaffected this tells us that g is in fact equal to the inverse function Um, conveniently this also tells us that f is equal to g inverse, right? So if a function is an inverse of another then you know, they're inversions of each other, right? f is g is g's inverse and g is f's inverse and one can verify That a function is an inverse of another by this function composition Let's take another let's look at another example a little bit more involved this time We have some raster functions in play here So we have the function f of x which equals 1 over x minus 1 and g of x which equals 1 over x and then add 1 to that To verify that these are inverse functions compose the two functions together so f composed with g evaluated at x So this means f of g of x and So g of x is going to be 1 over x plus 1 Insert that inside of the function f of x you get 1 over Replace that x with the 1 over x plus 1 and then minus 1 now We're going to try to simplify this thing in the denominator. You have 1 minus 1 that is cancel That'll leave you with 1 over 1 minus or 1 divided by x and whenever you divide by a fraction You're just going to multiply by the reciprocal so multiply by x over 1 and that simplifies just to be x the identity function Conversely if we do g of f of x Like so we're going to put inside of g the function f of x Well g of x is the function 1 over something plus 1 that something is now the function f of x Which itself is a rational function. So we have 1 over 1 over x minus 1 Plus 1 and like we saw a moment ago if you divide by a fraction you can multiply that by the reciprocal You have 1 times x minus 1 over 1 plus 1 So the the product will just become x minus 1, but then we have this plus 1 here still The plus 1 minus 1 they cancel and we end up with x again And this is what happens when you have an inverse function relationship when you do one into the other It just kind of unravels right each step along the way as you simplify these expressions. We're reforming inverse operations I'm multiplying to cancel out the division. I'm just adding to cancel out the subtraction that was in play here You can show that a function is an inverse of another right f inverse is just this function g You can show that two functions are in fact inverses of each other by composing them And if this gives you back x in both directions the inverse function property tells us that these are inverse functions Now you'll notice in this example We were checking if two functions were inverses this didn't actually show us how to compute Whether you know how to compute like what if I give you f what is the inverse of f? How do you have to pull that out of the air? Right? We'll talk about that in a later video at the moment I just want you to understand how one can verify two functions are inverses and thus Demonstrating us the inverse function property