in this video we're going to talk about how to show that two functions f of x and g of x are inverses of each other so let's say that f of x is equal to x squared plus five and g of x is the square root of x minus five are the two functions inverses of each other well let's see if they are so what we need to do is show that the composition of the two functions f of g of x is equal to x and also in the reverse order g of f of x is also equal to x if that's true then the two functions are inverses of each other so first let's determine f of g of x so g is on the inside of f so we're going to take this stuff and insert it into f so i'm going to replace x with the square root of x minus five so normally this would be x squared plus five that's the outside function f the inside function g i'm going to put it inside of f so i'm going to replace x with the square root of x minus 5. the square of the square root these cancel giving me what's inside which is x minus five and then if you add that to plus five negative five plus five is zero leaving behind x so f of g of x is equal to x now what about g of f of x so i'm going to take f and insert it into g so let's start with g it's the square root of x minus five so here is where x would be and i'm going to take this and replace it or substitute it for x so that's x squared plus five now 5 minus 5 is 0 so that leaves the square root of x squared and the square root of x squared is x so therefore these two functions are inverses of each other another way you can confirm the answer is by finding the inverse function of f f x and y are the same thing so to find the inverse function replace x with y next solve for y so i'm going to subtract both sides by five so x minus five is equal to the square root of y i mean y squared now you need to take the square root of both sides so the square root of x minus 5 is equal to y so therefore we could say that the inverse function of f is the square root of x minus 5 which is the same as g of x so f of x and g of x are indeed inverses of each other number two let's say that f of x is equal to three x plus eight and that g of x is equal to eight x squared minus three are these two functions inverses of each other well first let's determine f of g of x so let's start with the outside function f so it's three x plus eight but then let's replace x or substitute it with eight x squared minus three now let's distribute the three three times eight x squared is 24 x squared and then three times negative three is negative nine negative nine plus eight is negative one and we can't simplify this any further so notice that this does not equal to x so since what just happened here since f of g of x does not equal x then g is not the inverse of f and f is not the inverse of g so we could say g of x is not the inverse function of f and we could also say that let me make some space we could say that f of x is not the inverse function of g of x because this is not true now let's move on to another example so let's say that f of x is seven x plus five and that g of x is x minus five divided by seven are these two functions inverses of each other well let's start with f of g of x so f is going to be seven times x plus five and then we're going to take g and substitute it for x so this is gonna be x minus five divided by seven so we can see that seven divided by seven is one leaving behind x minus five and negative five plus five adds up to zero so we're left with x so f of g of x is equal to x so that's a good start now let's see what g of f of x is equal to so let's start with the outside function g so we have x minus five divided by seven now let's take f and insert it or substitute it into x so this is going to be seven x plus five and then minus five over seven so five minus five is zero and that leaves behind seven x over seven seven over seven is one so we get x so therefore we could say that f and g of x are inverses of each other so this means that f of x is the inverse of g of x and g of x is the inverse function of f of x so let's go ahead and find the inverse of f just another way to verify so let's say that y is equal to 7x plus five so we need to switch x and y if we subtract both sides by five we can see that seven x i mean x minus five equals seven y and then if we divide both sides by 7 we have x minus 5 over 7 is y so x minus 5 over 7 is the inverse function of f and we can see that this is the same as g of x so f of x and g of x are inverses of each other here's another problem that you could try for the sake of practice let's say that f of x is five over x plus three and that g of x is five divided by x minus three so determine whether or not if these two functions are inverses of each other so let's start with f of g of x so f is on the outside and let's replace x with g of x so i'm going to put 5 over x minus 3. so what do we need to do here we have like a complex fraction what i'm going to do is multiply the top and the bottom by x minus three so on the bottom these will cancel and so what i'm going to get is 5 times x minus 3 over 5. now notice that we can cancel five so then this gives us x minus three and negative three plus three adds up to zero so we get x so that's a good start so now let's focus on g of f of x so g is 5 divided by x minus three and we need to put f into g so f is five over x plus three so we can see that the threes will cancel and so we're left with five divided by 5 over x so i'm going to multiply the top and the bottom by x so these will cancel and this gives me 5 x over 5 so now i can cancel 5. leaving behind x so f of g of x is equal to x and g of f of x is equal to x so f of x and g of x are inverses of each other consider this example let's say that f of x is eight x plus seven and that g of x is seven over x minus eight so what is f of g of x and what's g of f of x are these two functions inverses of each other f and g well f is on the outside so let's put g inside of f so 8 times 7 over x 8 times 7 is 56 and then 8 times 8 is 64. now negative 64 plus 7 that's going to be negative 57 and as you can see this does not equal to x so f and g are not inverses of each other so here's the last example let's say that f is equal to x cubed plus five and that g of x is equal to the cube root of x minus five are they inverses of each other well let's start with f of g of x so the outside function f is x cubed plus 5. and let's insert g into f so let's replace x with the cube root of x minus five the cube root raised to the third power they will cancel leaving behind x minus five and then if we add five to that this will give us x and then for g of f of x g is on the outside so let's start with that and then that's x minus five and then insert f into g so once again the fives cancel five minus five is 0. so left with the cube root of x cubed in this case the 3's cancel this is the same as x raised to the 3 over 3 which is x to the 1 or x and so f and g are inverses of each other in this example you