 Welcome to the session. My name is Shashi and I am going to help you with the following question. The question says, find the relationship between A and B so that the function F defined by fx is equal to ax plus 1. If x is less than equal to 3, fx is equal to bx plus 3. If x is greater than 3, it is continuous at x is equal to 3. First of all, let us understand that if function F is continuous at x is equal to a, then left hand side limit of the function is equal to right hand side limit of the function is equal to value of the function at x is equal to a. So we can write x tending to a plus fx is equal to limit of x. tending to a minus fx is equal to f a. This is the right hand side limit of the function at x is equal to a. This is the left hand side limit of the function at x is equal to a. This is the value of the function at x is equal to a. All of them should be equal to make a function continuous. This is the key idea to solve the given question. Let us now start the solution. We are given fx is equal to ax plus 1 if x is less than equal to 3. And fx is equal to bx plus 3 if x is greater than 3. We are also given that function is continuous at x is equal to 3. So we can write function f is continuous at x is equal to 3. This is given in the question. Now find out right hand side limit of the function at x is equal to 3. So we can write limit of x tending to 3 plus fx is equal to limit of x tending to 3 plus. bx plus 3 this is equal to 3b plus 3. So right hand side limit of the function at x is equal to 3 is equal to 3b plus 3. Let us now find out left hand side limit of the function at x is equal to 3. So we can write limit of x tending to 3 minus fx is equal to limit of x tending to 3 minus. ax plus 1 this is equal to 3a plus 1. Now we know given function f is continuous at x is equal to 3. So these two limits that is right hand side limit and left hand side limit should coincide each other. So we can write 3a plus 1 is equal to 3b plus 3. This implies 3a is equal to 3b plus 3 minus 1. This implies 3a is equal to 3b plus 2. Now dividing both sides by 3 we get a is equal to b plus 2 upon 3. So the required relationship between a and b is a is equal to b plus 2 upon 3. This is our required answer. This completes the session. Goodbye.