If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Verifying inverse functions by composition

CCSS.Math:

## Video transcript

let's say that f of X is equal to X plus seven to the third power minus one and let's say that G of X G of X is equal to the cube root of x plus one the cube root of x plus one minus seven now what I want to do now is evaluate F of F of G of X I want to evaluate F of G of X and I also want to evaluate G of f of X G of f of X and see what I get and I encourage you like always pause the video and try it out alright let's first evaluate f of G of X so that means G of X this expression is going to be our input so everywhere where we see an X in the definition for f of X we would replace it with all of G of X so f of G of X is going to be equal to so it's going to be equal to well I see an X right over there so I'd write all of G of X there so that's the cube root of x plus 1 minus 7 and then I have plus 7 plus 7 to the third power minus 1 notice wherever I saw the X since I'm taking F of G of X I replace it with what G of X is and so that is the cube root of x plus 1 minus 7 all right now let's see if we can simplify this well we have a minus 7 plus 7 so that simplifies nicely so this just becomes this is equal to I can do it a neutral color now this is equal to the cube root of x plus 1 to the third power minus 1 well if I take the cube root of x plus 1 and then I raise it to the third power well that's just going to give me X plus 1 so this part this part just simplifies to x plus 1 and then I subtract 1 so it all simplified out to just being equal to X so we're just left with an X so f of G of X is just X so now let's try what G of f of X is so G of f of X is going to be equal to I'll do it right over here this is going to be equal to the cube root of and actually I could let me write it out wherever I see an X I can write f of X instead I didn't do it that last time I went directly and replace with the definition of f of X but just to make it clear what I'm doing so everywhere I'm seeing an X I replace it with an f of X so the cube root of f of X plus 1 minus 7 well that's going to be equal to the cube root of cube root of f of X which is all of this business over here so that is X plus 7 to the third power minus 1 and then we add 1 and we add 1 and then we subtract the 7 lucky for us this is subtracting 1 and adding 1 those cancel out and so we're going to take the cube root of x plus 7 to the third power well the cube root of x plus 7 to the third power is just going to be X plus 7 so this is going to be X plus 7 so all of this business simplifies to X plus 7 and then we do subtract 7 and these two cancel out or they negate each other and we are just left with X so we see something very interesting F of G of X is just X and G of f of X is X so if we in this case if we start with an X if we start with with an X we input it into the function G and we get G of X we get G of X and then we input that into the function f and we put that into the function f F of G of X gets us back to X it gets us back to X so we kind of did a round trip and the same thing is happening over here if I put X into f of X if I put X into F of sorry I get put X into the function f and I get f of X the output is f of X and then I input that into G into the function G into the function G once again I do this round-trip and I get back to X another way to think about it another way to think about it if you view this as so this is these are these are both composite functions but one way to think about it is if these are the set of all possible inputs into either of these composite functions and then these are the outputs so you are starting with an X you are starting with an X I'll do this case first so G is a mapping let me let me write down so G is going to map be a mapping from X to G of X so this is what G is doing so the function G maps from X to some value G of X G of X and then if you were to apply F to this value right over here if you apply F to this value to G of X you get all the way back to X so that is f of G of X and vice-versa if you go if you start with X and you apply f of X first so if you start with F if you apply f of X first let me do that so if you apply f of X first you say you get to this value so that is f of X so you applied the function f but you apply the function G to that you apply the function G to that you get back so this is G of G of f of X I should say G of or G of F where we're applying the function G to the value f of X and so it eats since we get a round trip either way we know that the functions G and F are inverses of each other in fact we can write we can write that f of X is equal to the inverse of G of X inverse of G of X and vice versa G of X is equal to the inverse of f of X inverse of f of X hope you enjoyed that