 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that Find the domain of the following functions. First part f of x is equal to fourth root of 2x plus 5 the whole And second part f of x is equal to fifth root of x plus 6 the whole Let us start with the solution of the given question Here in the first part we are given a function f of x is equal to fourth root of 2x plus 5 the whole We have to find its domain. To find the domain of the given function we shall follow some steps And here the first step is to determine its index and radicant Here index will be equal to fourth and radicant will be given by 2x plus 5 Now since index is an even number so we must restrict the domain to make radicant that is expression inside the root greater than equal to 0 So in the second step as index is equal to fourth which is an even number so we set radicant greater than equal to 0 Hence 2x plus 5 is greater than equal to 0 Now in the next step we shall solve this inequality Now subtracting 5 from both the sides we get 2x plus 5 minus 5 is greater than equal to 0 minus 5 Which implies that 2x is greater than equal to minus 5 Now dividing both sides by 2 we get x is greater than equal to minus 5 by 2 Now in the next step we shall write the answer in interval or set notation So in set notation domain of the function f of x is equal to set of all x such that x is greater than or equal to minus 5 by 2 where x belongs to the set of real numbers And in interval notation domain of f of x is equal to the semi-closed interval minus 5 by 2 to infinity Now this is the required answer to the first part of the question Now we move on to the second part of the question Here we have to determine the domain of the function f of x is equal to fifth root of x plus 6 the whole Here we should note that index is equal to 5 and radicant is equal to x plus 6 And here we see that the index is an odd number We know that if index is an odd number then domain is the set of all real numbers As here the index is an odd number So in set notation domain of the function f of x is equal to set of all x such that x belongs to the set of real numbers And in interval notation domain of the function f of x is equal to the open interval minus infinity to infinity This is the required answer to the second part of the question This completes our session. Hope you enjoyed this session