 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that Examine the end behavior of the function where the function given is f of x is equal to minus 4 x raised to power 5 plus 2 x raised to power 3 plus 2 x squared plus 1. Let us start with the solution of the given question. We are given a polynomial function f of x is equal to minus 4 x raised to power 5 plus 2 x raised to power 3 plus 2 x squared plus 1. We know that for n greater than or equal to 0 we have the polynomial f of x is equal to a n x raised to power n plus a n minus 1 x raised to power n minus 1 plus and so on plus a 2 x squared plus a 1 x plus a 0. Then the polynomial given by its leading term g of x is equal to a n x raised to power n has same end behavior as f of x. So, let us see the leading term of the given polynomial and that is given by minus 4 x raised to power 5. So, to know the end behavior of f of x we examine its leading term here the leading coefficient is negative that is minus 4 and the highest power is 5 which is odd. So, as x will become very large the value of f of x will become very small as x will become very small the value of f of x will become very large. So, its end behavior is f of x tends to minus infinity as x tends to infinity and f of x tends to infinity as x tends to minus infinity. If we see the graph of the function f of x is equal to minus 4 x raised to power 5 plus 2 x raised to power 3 plus 2 x squared plus 1. We see that its branch lying in second quadrant shows that as x tends to minus infinity that is to the smaller negative values the value of the function f of x becomes greater and greater and it approaches to plus infinity. Similarly, its other branch is moving downwards in fourth quadrant which shows that as x tends to plus infinity that is to the positive values the value of the function f of x becomes smaller and smaller and it approaches to minus infinity. This is the required answer. This completes our session. Hope you enjoyed this session.