 Hi and welcome to the session. Let us solve the following question. Consider function f from r plus to interval minus 5 infinity which is closed at minus 5 and open at infinity given by fx is equal to 9x square plus 6x minus 5. Show that f is invertible with f inverse y is equal to under root y plus 6 minus 1 upon 3. 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 gof is equal to identity function on x and f4g is equal to identity function on y. Then g is called the inverse of function f and it is denoted by f inverse. This is the t idea to solve the given question. Let us now start the solution. We are given function f from r plus to interval minus 5 infinity which is closed at minus 5 and open at infinity. It is given by fx equal to 9x square plus 6x minus 5. Let us consider any arbitrary element y in range of function f. Then y must be equal to fx for some x in r plus. Now we know fx is equal to 9x square plus 6x minus 5. So we can write y is equal to 9x square plus 6x minus 5. Now adding and subtracting 1 in this expression we get y is equal to 3x plus 1 whole square minus 6. This implies y plus 6 is equal to 3x plus 1 whole square. This implies under root of y plus 6 is equal to 3x plus 1. This implies x is equal to under root of y plus 6 minus 1 upon 3. This gives the function g from interval minus 5 infinity which is closed at minus 5 and open at infinity 2 r plus and is defined by gy equal to under root of y plus 6 minus 1 upon 3. Let us now find out g of fx. g of fx is equal to g of 3x plus 1 whole square minus 6. Now this is further equal to under root of 3x plus 1 whole square minus 6 plus 6 minus 1 upon 3. Now minus 6 and plus 6 will get cancelled and we get 3x plus 1 minus 1 upon 3. We have neglected the all negative values as x belongs to positive real numbers. So we get plus 1 and minus 1 will get cancelled. We get 3x upon 3. Now 3 and 3 will get cancelled and we get x which is equal to identity function on. Now let us find out f o gy. f o gy is equal to f gy. Now we know gy is equal to under root of y plus 6 minus 1 upon 3. So we can write under root of y plus 6 minus 1 upon 3 f of under root of y plus 6 minus 1 upon 3 is equal to 3 multiplied by under root y plus 6 minus 1 upon 3 plus 1 whole square minus 6. Now simplifying we get 3 and 3 will get cancelled. This is equal to under root of y plus 6 minus 1 plus 1 whole square minus 6. This is further equal to under root of y plus 6 whole square minus 6 minus 1 and plus 1 will get cancelled. Now this is further equal to y plus 6 minus 6. Here we have neglected all the negative values as y is greater than minus 5. Now plus 6 and minus 6 will get cancelled and we get y that is the identity function on interval minus 5 infinity which is closed at minus 5 and open at infinity. Now gofx is equal to identity function on r plus and f o gy is equal to identity function on the interval minus 5 infinity which is closed at minus 5 and open at infinity implies that f is invertible or we can say function f is invertible and g is the inverse of function f. So we can write this implies f is invertible and g is equal to f inverse. Inverse of f is given by gy equal to under root of y plus 6 minus 1 upon 3. So we can write f inverse y is equal to under root of y plus 6 minus 1 upon 3. Hence proved. This is our required answer. This completes the session. Goodbye.