 Hello and welcome to the session. In this session, we are going to discuss the following question and the question says that. Restrict the domain of f of x is equal to x minus one whole square, so its inverse exists. Also find its inverse. Now we know that inverse of a function y is equal to f of x exists if the function is one to one. This can be known as if rightly that is we find its inverse and if the plus minus sign exists in its inverse then it is not one to one function and to make it one to one function, we restrict domain of f of x with this key idea we proceed to the solution. Now we are given a function f of x is equal to x minus one whole square and from the key idea we know that inverse of a function y is equal to f of x exists if the function is one to one. Now its inverse will exist if it is one to one function that is when we find its inverse then value of y should not be in terms of plus minus sign because it indicates that for one value of y there corresponds two values of x so inverse found will not be a function. So let us find its inverse here domain is given by x which belongs to the set of real numbers. Now in this function we replace f of x by y and we get y is equal to x minus one whole square. Now we switch x and y and we get x is equal to y minus one whole square. Now we solve this equation for y and we get plus minus of square root of x is equal to y minus one that is taking square roots on both sides of the equation we get plus minus square root of x is equal to y minus one which implies that one plus minus square root of x is equal to y or we can also write it as y is equal to one plus minus square root of x. Now we replace y by f inverse of x so we have f inverse of x is equal to one plus minus square root of x so it is not one one because it is having two values of x corresponding to the same value of y and here domain is all real numbers. Now if we want to find its inverse then we will have to make it one to one function by restricting its domain. Let us see its graph. Now here is the graph of the equation y is equal to x minus one whole square it is an upward parabola with vertex one zero. Now here we see that for each y there are two corresponding values of x here for point with coordinates zero one and point with coordinates two one we see that for same value of y that is for one we have two values of x that is x is equal to zero and x is equal to two so it is not one to one function but instead of taking domain as x belonging to all real numbers if we take only those values of x which are greater than or equal to one we have only this remaining curve now see for each value of y there is exactly one value of x so now the function has become one to one so we restrict the domain to x is greater than or equal to one or we can write it as x minus one is greater than equal to zero. Now for x is greater than or equal to one we will find its inverse using the same procedure here we have y is equal to x minus one whole square for x is greater than or equal to one now we switch x and y and we get x is equal to y minus one whole square now we shall solve this equation for y and we get square root of x is equal to y minus one since we have taken the value of x as greater than or equal to one so we will not take value of y minus one as plus minus square root of x we have taken only positive square root of x it further implies that square root of x plus one is equal to y or we can write it as y is equal to square root of x plus one so now we have f inverse of x is equal to square root of x plus one which is the inverse of f of x is equal to x minus one whole square for x is greater than or equal to one which is the required answer this completes our session hope you enjoyed this session