 Hello and welcome to the session. In this session we discussed the following question which says what is the domain and range of fx equal to x to the power 4 upon 1 minus x to the power 4. Is f 1 to 1? Let's move on to the solution now. We are given the function fx equal to x to the power 4 upon 1 minus x to the power 4. Now, we can easily see that x to the power 4 upon 1 minus x to the power 4 is not defined when 1 minus x to the power 4 that is the denominator is equal to 0. Or you can say that is when we have x equal to plus minus 1. So when we have x equal to plus minus 1, x to the power 4 upon 1 minus x to the power 4 is not defined. And in that case we would say that the domain of the function f is equal to the set of real numbers are minus the set with elements minus 1, 1. That is the set of real numbers excluding minus 1 and 1 is the domain of the function f. So we have got the domain of the function f. Now to find the range of the function f, we take let y be equal to x to the power 4 upon 1 minus x to the power 4. So this means 1 plus y is equal to 1 plus x to the power 4 upon 1 minus x to the power 4. So further we have 1 plus y is equal to 1 upon 1 minus x to the power 4. So from here we get y is equal to x to the power 4 into 1 plus y. Here in place of 1 upon 1 minus x to the power 4 we have taken 1 plus y. So we get y equal to x to the power 4 into 1 plus y. This further gives us x to the power 4 is equal to y upon 1 plus y. From here we have x is equal to fourth root of y upon 1 plus y. So now for x to be real we must have y upon 1 plus y to be greater than equal to 0 and also 1 plus y should not be equal to 0. Therefore we say that the range of the function f is equal to y belongs to r such that y is greater than equal to 0. So this is the range of the function f. Now we have to check whether the function f is 1 to 1 or not. We have the function fx equal to x to the power 4 upon 1 minus x to the power 4. Now for the function f which goes from say a to b is 1 to 1 if f of x1 equal to f of x2 implies that x1 is equal to x2 where we have x1 x2 belongs to the set a. We take let x1 be equal to minus 2 and x2 be equal to 2. Let's find out f of x1 that is f of minus 2 this is equal to minus 2 to the power 4 upon 1 minus minus 2 to the power 4. This is equal to 16 upon 1 minus 16. So this is equal to minus 16 upon 15 that is we get f of minus 2 is equal to minus 16 upon 15. Now let's find out f of x2 which is f of 2 this is equal to 2 to the power 4 upon 1 minus 2 to the power 4. So this is equal to 16 upon 1 minus 16 which is again equal to minus 16 upon 15. Thus we get f of 2 is equal to minus 16 upon 15. Therefore as you can see that f of minus 2 is equal to f of 2 but minus 2 is obviously not equal to 2. Therefore the condition of the function f to be 1 to 1 is not satisfied and we say that the function fx equal to x to the power 4 upon 1 minus x to the power 4 is not 1 to 1. So this completes the session hope you have understood the solution of this question.