 Hello and welcome to the session. In this session we discussed the following question which says, suppose r plus is the set of all non-negative real numbers, prove that the function f which goes from r plus to closed interval minus 10 infinity defined by fx equal to 4x square plus 16x minus 2 is invertible. Before we move on to the solution, let's see when a function is set to be invertible. Consider a function f which goes from x to y, this function f is invertible if and only if f is 1 1 and on 2. This is the key idea that we use for this question. Now let's proceed with the solution. We have a function f which goes from r plus to the closed interval minus 10 infinity and it is defined as fx is equal to 4x square plus 16x minus 2. We have to show that f is invertible. So for this we have to show that f is 1 1 and on 2. First we will show that f is 1 1. First of all we take let f of x1 be equal to f of x2 where we have x1 and x2 belongs to r plus. So this would mean that 4x1 square plus 16x1 minus 2 is equal to 4x2 square plus 16x2 minus 2. Further we have 2x1 plus 4 the whole square minus 14 is equal to 2x2 plus 4 the whole square minus 14 or you can say we get 2x1 plus 4 whole square is equal to 2x2 plus 4 whole square 4 is equal to which gives us 2x1 is equal to 2x2 that is we get x1 is equal to x2. If x1 is equal to f of x2 this implies that x1 is equal to x2 and therefore we say that fx is 1 1 so the function is a 1 1 function. Now next we have to show that f is an on 2 function this we take let y be equal to x minus 2 this means that y is equal to 2x plus 4 the whole square minus 14 or y plus 14 is equal to 2x plus 4 whole square whole is equal to square root of y plus 14 or you can say that x plus 2 is equal to square root of y plus 14 upon 2 this means that we get x is equal to square root of y plus 14 upon 2 minus 2 this means that x is equal to square root of y plus 14 minus 4 whole upon 2 y belongs to the closed interval minus 10 infinity there exists x is equal to square root of y plus 14 minus 4 whole upon 2 and r plus is equal to f of y plus 14 minus 4 whole upon 2 and this is further equal to 2 into square root of y plus 14 minus 4 whole upon 2 the whole square minus 14 that is we get this is equal to square root of y plus 14 minus 4 whole square minus 14 or you can say this is equal to square root of y plus 14 whole square minus 14 this is equal to y plus 14 minus 14 this 14 and minus 14 cancels and this is equal to y that is we get fx is equal to y and therefore we say that f is on 2 since we get therefore we say that f is invertible now let's find out f inverse is equal to x which means that y is equal to 4x square plus 16x minus 2 or you can say that y is equal to 2x plus 4 the whole square minus 14 as here from the value for y we got the value for x so when we had y is equal to 4x square plus 16x minus 2 or y is equal to 2x plus 4 the whole square minus 14 we get the value for x as square root of y plus 14 minus 4 this whole upon 2 and this x is f inverse y so we get f inverse y is equal to square root of y plus 14 minus 4 whole upon 2 the function f inverse which goes from the closed interval minus 10 infinity to r plus such that we have f inverse y is equal to square root of y plus 14 minus 4 whole upon 2 so we have got the value for f inverse also and we know that f is invertible so with this we complete the session hope you have understood the solution of this question