 Hi and welcome to the session. Let us discuss the following question. Question is, consider function f from r to r where r is the set of real numbers. Given y, fx is equal to 4x plus 3. Show that f is invertible. Find the inverse of f. First upon let us understand that a function f from x to y is said to be invertible if there exists a function g from y to x such that g of f is equal to ix that is the identity function on x f for g is equal to iy that is the identity function on y. Then function g is called the inverse of f and is denoted by f inverse. So, we can write this is the k idea to solve the given question. Let us now start the solution. We are given function f from r to r given y fx is equal to 4x plus 3. Let us consider any arbitrary element y in range f. y must be equal to fx for some x in domain r. Now y is equal to fx. We can write y is equal to 4x plus 3. We know fx is equal to 4x plus 3. So, we can write y is equal to 4x plus 3. This implies 4x is equal to y minus 3. This further implies x is equal to y minus 3 upon 4. Now this gives the function g given by g of y equal to y minus 3 upon 4. Now let us find out g of fx. g of fx is equal to g of fx. We know fx is equal to 4x plus 3. So, we get g of 4x plus 3. Now this is further equal to 4x plus 3 minus 3 upon 4. We know g of y is equal to y minus 3 upon 4. So, g of 4x plus 3 would be equal to 4x plus 3 minus 3 upon 4. Plus 3 and minus 3 will get cancelled and we get 4x upon 4. 4 and 4 will get cancelled and we get x. That is it is equal to identity element on the function r where r is the set of all real numbers. So, we get g of fx is equal to identity function on r. Now let us find out f for g y. f for g y is equal to f g y. This is equal to f of y minus 3 upon 4. This is further equal to 4 multiplied by y minus 3 upon 4 plus 3. We know fx is equal to 4x plus 3. So, we will substitute for x y minus 3 upon 4. Now on simplifying we get 4 y minus 12 plus 12 upon 4. Now this is equal to minus 12 plus 12 will get cancelled. We get 4 y upon 4. Now 4 and 4 will get cancelled. We get f for g y is equal to y which is equal to identity function on r. Now we can see g of fx is equal to identity function on r and f for g y is also equal to identity function on r. So, therefore function f is invertible. So, we can write this implies f is invertible and g is the inverse of f. So, our final answer is f inverse is given by f inverse high. You know we can represent g by f inverse. So, f inverse y is equal to y minus 3 upon 4. This is our required answer. This completes the session. Hope you enjoyed the session. Goodbye.