 Hi and welcome to the session. Let us discuss the following question. Question says, let F, T and H be functions from R to R, where R is the set of real numbers. Show that F plus G OH is equal to FOH plus GOH. Then another part is F multiplied by G OH is equal to FOH multiplied by GOH. First of all, let us understand that if we are given a function F from A to B and there is another function G from B to C, then their composition is a function GOF from A to C given by GOFX is equal to GFX for every X belonging to F. This is the key idea to solve the given question. Let us now start the solution. We are given functions F, G and H are from R to R. So we can write F, G and H are functions R to R. So therefore, their composition F plus GOH, FOH, GOH, F multiplied by GOH are all our functions from R to R. So first of all, we will start with the first part. For this, we have to show F plus G OH is equal to FOH plus GOH. Let us start the proof now. We can define the function F plus GOH from R to R. As we know, F, G and H are functions from R to R. So their composition would also be a function from R to R. Similarly, we can write FOH plus GOH is also a function from R to R. Now, for any X belonging to real numbers, we can write F plus G OH is equal to F plus GHX using key idea. We know GOFX is equal to GFX. So we get F plus GOHX is equal to F plus G HX. Now, this is further equal to FHX plus GHX can be written as FOHX plus GOHX. So we get F plus GOHX is equal to FOHX plus GOHX for all X belonging to set of real numbers. Or we can write it as F plus GOH is equal to FOH plus GOH. This is the required answer. This completes the first part of the question. Let us now start with the second part of the question. That is, we have to show F multiplied by G OH is equal to FOH multiplied by GOH. Let us now start with the proof of the second part. You know, F, G and H are the functions from R to R. So their composition F multiplied by G OH is also a function from R to R, FOH multiplied by GOH is also a function from R to R. Now, for any X belonging to set of real numbers, we can write F multiplied by G OH is equal to F multiplied by G HX. Using key idea, we know GOFX is equal to G of FX. So, what we have done here is F multiplied by GOHX is equal to F multiplied by G HX. Now, this is further equal to F of HX multiplied by G of HX. Now, this is further equal to FOHX multiplied by GOHX. This can be written as FOH multiplied by GOH. For any X belonging to R, F multiplied by G OH is equal to FOH multiplied by GOHX. Or we can write it as F multiplied by G OH is equal to FOH multiplied by GOH. You are required to prove this only. So, this completes the session. Hope you understood the session. Take care and goodbye.