 Hi and welcome to the session. Let us discuss the following question. Question is, consider function f from r plus to interval 4 infinity which is closed at 4 and open at infinity given by fx is equal to x square plus 4. Show that f is invertible with the inverse f inverse of f given by f inverse y is equal to under root of y minus 4 where r plus is the set of all non-negative real numbers. First of all let us understand that function f from x to y is said to be invertible if there exists a function g from y to x such that 0f is equal to identity function on x f o g is equal to identity function on y. Then g is called the inverse of f it is denoted by f inverse. This is the key idea to solve the given question. Let us now start the solution. We are given function f from r plus that is the set of all positive real numbers to interval 4 infinity which is closed at 4 and open at infinity it is given by fx is equal to x square plus 4. Now let y be any arbitrary element in range of function f then y must be equal to fx for some x in domain r plus. Now we know fx is equal to x square plus 4. So we will substitute for fx x square plus 4. So we get y is equal to x square plus 4. This implies x square is equal to y minus 4. This implies x is equal to under root of y minus 4. This gives function g from interval minus 4 infinity which is closed at 4 and open at infinity to r plus defined by g y equal to under root of y minus 4. Let us now find out g of fx. We know g of fx is equal to g fx fx is equal to x square plus 4. Now g of x square plus 4 is equal to under root of x square plus 4 minus 4. Plus 4 and minus 4 will get cancelled and we get g of fx equal to x. We will reflect the negative value as x belongs to set of positive real numbers. Let us now find out f o g y f o g y is equal to f g y. Now we know g y is equal to under root of y minus 4. So we can write f of under root of y minus 4. Now this is further equal to under root of y minus 4 whole square. This is equal to y minus 4 plus 4. Since x is an element of set of positive real numbers. So we have neglected all the negative values here. Now y minus 4 plus 4 is equal to y as minus 4 and plus 4 will cancel each other. Now this is equal to identity function on the interval 4 infinity which is closed at 4 and open at infinity. And also g o fx is equal to x that is it is equal to identity function on r plus. g o fx is equal to identity function on r plus and f o g y is equal to identity function on the interval 4 infinity which is closed at 4 and open at infinity implies that g is the inverse of function f. So we can write this implies f is invertible e is the inverse of f that is f inverse is equal to g. We know g y is equal to under root y minus 4. So this implies f inverse y is equal to under root of y minus 4. This completes the session. Hope you understood the session. Goodbye.