 Hello and welcome to the session. In this session we will discuss a question which says that find the values of f of minus 1 by 2 and f of 2x plus 1 the whole, then f of x is equal to 2 into x plus 3 whole square and second part is if p of t is the population of a city after t years from the year 2000, interpret this statement p of 0 is equal to 456,300 and p of 10 minus p of 9 is equal to 4500. Now let us start with the solution of the given question. In the first part we are given a function f of x is equal to 2 into x plus 3 whole square and firstly we have to find f of minus 1 by 2. So there let us put x is equal to minus 1 by 2 in f of x and here we will have f of minus 1 by 2 is equal to 2 into minus 1 by 2 plus 3 whole square. Now here we will have 2 into, now taking LCD it will be minus 1 plus 6 whole upon 2 whole square and this is equal to 2 into, now minus 1 plus 6 is 5 upon 2 whole square which is equal to 2 into, now 5 square is 25 upon 2 square is 4, now 2 into 2 is 4 so this is equal to 25 upon 2 therefore f of minus 1 by 2 is equal to 25 upon 2 and now we have to find f of 2x plus 1 the whole. For this we will replace x by 2x plus 1 in f of x and here we have f of 2x plus 1 the whole is equal to 2 into plus 1 the whole plus 3 whole square further this is equal to 2 into 2x plus 1 plus 3 whole square which is equal to 2 into 2x plus now 1 plus 3 is 4 whole square. Now using this identity that is a plus v whole square is equal to a square plus 2ab plus b square this will be equal to 2 into, now let us denote 2x by a and 4 by b so this is a plus b whole square that is 2x plus 4 whole square which will be 2x whole square plus 2ab that is 2x into b that is 4 plus b square that is 4 square the whole further this is equal to 2 into 4x square plus now 2 into 2 into 4 is 16 into x that is 16x plus 4 square is 16 the whole further this is equal to now 2 into 4x square is 8x square plus 2 into 16x is 32x plus 2 into 16 is 32 so f of 2x plus 1 the whole is equal to 8x square plus 32x plus 32. Now let us start with the second part now here we are given p of t is the population of a city after t years from the year 2000 that means we have started calculating population after the year 2000 also we are given p of 0 is equal to 456300 it means value of t is equal to 0 now t is equal to 0 means after 0 years the population of city is 456300 initial population that is population of the city in the year 2000 is 1300 year given minus p of 9 is equal to 4500 that p of 10 means population from the year 2000 that is the year 2010 p of 9 means population after 9 years from the year 2000 that is the year 2009 and here negative sign notes difference of 100 of the given question session hope you all have enjoyed the session.