 Hi and welcome to the session. Let us discuss the following question. Question says, find all the points of the discontinuity of function f, where function f is defined by fx is equal to modulus of x plus 3, if x is less than equal to minus 3, fx is equal to minus 2x, if x is greater than minus 3 and less than 3, fx is equal to 6x plus 2, if x is greater than equal to 3. First of all let us understand that function f is discontinuous that x is equal to a, if left hand side limit of the function is not equal to right hand side limit of the function at x is equal to a, that is limit of x tending to a minus fx is not equal to limit of x tending to a plus fx. This is the key idea to solve the given question. Let us now start with the solution. We know fx is equal to modulus of x plus 3, if x is less than equal to minus 3, fx is equal to minus 2x, if x is greater than minus 3 and less than 3, fx is equal to 6x plus 2, if x is greater than equal to 3. Now clearly we can see function f is defined at all the values less than equal to minus 3, greater than equal to 3 and all the values between minus 3 and minus 3. So we can say function f is defined at all the real numbers. First of all let us discuss the continuity of the function at all the values less than minus 3. We know fx is equal to modulus of x plus 3, if x is less than equal to minus 3. Now this is a modulus function and we know modulus function is continuous at every real number. This is a constant function, constant function is also continuous at every real number. Now sum of two continuous functions is also continuous. So we can write function f is continuous at every real number less than minus 3. Let us discuss the continuity of the function for all the values between minus 3 and 3. We know fx is equal to minus 2x, if x is greater than minus 3 and less than 3. Now this is a polynomial function and we know polynomial function is continuous at every real number. So given function is continuous at every real number between minus 3 and 3, we can write function f is continuous at every real number between minus 3 and 3. Let us discuss the continuity of the function at all the values of x greater than 3. We know fx is equal to 6x plus 2 if x is greater than equal to 3. Now this is a polynomial function and we know polynomial function is continuous at every real number. So given function f is continuous at all the real values greater than 3. So we can write function f is continuous at every real number greater than 3. Now let us discuss continuity of the function fx is equal to minus 3. Now we know fx is equal to modulus of x plus 3 if x is less than equal to minus 3. So first of all let us find out value of the function fx is equal to minus 3 that is equal to modulus of minus 3 plus 3. Now this is further equal to 3 plus 3 which can be written as 6. So we get value of the function at x is equal to minus 3 is equal to 6. Now we will find out left hand side limit of the function at x is equal to minus 3. So we can write limit of x tending to minus 3 minus fx is equal to limit of x tending to minus 3 minus modulus of x plus 3 which is further equal to 3 plus 3 or we can simply write it as 6. Now let us find out right hand side limit of the function at x is equal to minus 3 we know fx is equal to minus 2x if x is greater than minus 3. So we can write limit of x tending to minus 3 plus fx is equal to limit of x tending to minus 3 plus minus 2x which can be further written as minus 2 multiplied by minus 3. Now this is equal to 6 clearly we can see value of the function is equal to left hand side limit of the function is equal to right hand side limit of the function at x is equal to minus 3. So this implies function f is continuous at x is equal to minus 3. Now let us take continuity of the function at x is equal to 3. Now we know fx is equal to minus 2x if x is less than 3 and fx is equal to 6x plus 2 if x is greater than equal to 3. So first of all let us find out value of the function at x is equal to 3. Now this is equal to 6 multiplied by 3 plus 2 which can be further written as 18 plus 2. So we get f3 is equal to 20. Now let us find out left hand side limit of the function at x is equal to 3 that is limit of x tending to 3 minus fx is equal to limit of x tending to 3 minus minus 2x. We know if x is less than 3 fx is equal to minus 2x. Now this is further equal to minus 2 multiplied by 3 which can be further written as minus 6. So we get left hand side limit of the function at x is equal to 3 is equal to minus 6. Now let us 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 6x plus 2. We know fx is equal to 6x plus 2 if x is greater than equal to 3. Now this limit is equal to 6 multiplied by 3 plus 2 which can be further written as 20. Now clearly we can see left hand side limit of the function is not equal to right hand side limit of the function at x is equal to 3. So the given function is discontinuous at x is equal to 3. So our required answer is given function f is discontinuous at x is equal to 3. This completes the session. Hope you understood the session. Take care and have a nice day.