 Hello and welcome to the session. In this session we discussed the following question which says, prove that the function f which goes from r to r given by fx equal to cos x is neither one to one nor on two. First let's recall the conditions for a function to be one to one and on two given a function f which goes from a to b then we say f is one to one if f of x1 is equal to f of x2 implies that x1 is equal to x2 where we have x1 x2 are the elements of set A then the function f is set to be on two if for each y belongs to the set B there exist at least one element x which belongs to the set A such that y is equal to fx also we have one more condition for a function f to be on two the function f is on two if and only if the range of the function f is equal to the set B this is the key idea that we use for this question. Let's move on to the solution now we are given a function f which goes from r to r and the function f is given by fx equal to cos x we have to show that the function f is neither one nor on two so first of all let's show that f is not one to one we know that f is one to one if f of x1 is equal to f of x2 implies that x1 is equal to x2 and x1 x2 are the elements of set A so we take let x1 be equal to zero and x2 be equal to pi now f of x1 would be equal to f of zero and this would be equal to cos zero since we have fx equal to cos x and this is equal to one so f of x2 is equal to f of pi which is equal to cos pi and this is also equal to one thus we get f of x1 is equal to f of x2 that is f of zero is equal to f of pi but zero is not equal to pi that is we have f of x1 is equal to f of x2 but x1 is not equal to x2 therefore we say that the function f is not one to one so we have shown that f is not one to one now next we show that f is not on two we have the function fx is equal to cos x so in the key idea we had given one condition that is f is on two if and only if range of the function f is equal to the set B so in this case f is a function which goes from R to R so to show that f is one one we should have that range of the function f is the set R but here range of the function f which is cos x is equal to the closed interval minus one one which is a subset of R and it is not R therefore we say that the function f is not on two since it ranges a subset of the set R thus we have the function fx equal to cos x is neither one to one nor on two so this completes the session hope you have understood the solution of this question