If you divide (6x +3) by (x +1) you get some quotient and some remainder. You can do it a variety of ways, including synthetic division and long division. The method used here is to rewrite 6x+3 as a multiple of x+1 with some constant term added. .. 6x +3 = (6x +3) +3 -3 .. = (6x +3 +3) -3 .. = (6x +6) -3 .. = 6(x +1) -3
Now, you can divide this by (x +1) and you have [tex]\frac{6x+3}{x+1} = \frac{6(x+1)-3}{x+1} = \frac{6(x+1)}{x+1} +\frac{-3}{x+1} = \frac{-3}{x+1}+6[/tex] Then the boxes can be filled from ... [tex]g(x)=\frac{6(x+1)+(-3)}{x+1}=\frac{-3}{x+1}+6[/tex]
You know that .. f(x) +6 represents a translation of f(x) by 6 units up And you know that .. f(x +1) represents a translation of f(x) by 1 unit left
So, you can figure that .. g(x) = f(x +1) +6 will represent a translation of 1 unit left and 6 units up of f(x) = -3/x.