Function f is defined by f x = x + 6 x + 5 x + 1 . Function g

Question

Function f is defined by f x = x + 6 x + 5 x + 1 . Function g is defined by g x = f x - 1 . The graph of y = g x in the xy -plane has x -intercepts at a , 0 , b , 0 , and c , 0 , where a , b , and c are distinct constants. What is the value of a + b + c ?

Accepted Answer

Choice B is correct. It's given that g x = f x - 1 . Since f x = x + 6 x + 5 x + 1 , it follows that f x - 1 = x - 1 + 6 x - 1 + 5 x - 1 + 1 . Combining like terms yields f x - 1 = x + 5 x + 4 x . Therefore, g x = x x + 5 x + 4 . The x -intercepts of a graph in the xy -plane are the points where y = 0 . The x -coordinates of the x -intercepts of the graph of y = g x in the xy -plane can be found by solving the equation 0 = x x + 5 x + 4 . Applying the zero product property to this equation yields three equations: x = 0 , x + 5 = 0 , and x + 4 = 0 . Solving each of these equations for x yields x = 0 , x = - 5 , and x = - 4 , respectively. Therefore, the x -intercepts of the graph of y = g x are 0 , 0 , - 5 , 0 , and - 4 , 0 . It follows that the values of a , b , and c are 0 , - 5 , and - 4 . Thus, the value of a + b + c is 0 + - 5 + - 4 , which is equal to - 9 . Choice A is incorrect. This is the value of a + b + c if g x = f x + 1 . Choice C is incorrect. This is the value of a + b + c - 1 if g x = x - 6 x - 5 x - 1 . Choice D is incorrect. This is the value of a + b + c if f x = x - 6 x - 5 x - 1 .

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