O Graphs and Functions 3/5 Composition of two rational functions X 11 For the functions f (x) = x+ 2 and g (x) = , find the composition fog and simplify your answer as much as possible. Write the domain using X interval notation. VO ( f . =) (x) = 0 0|01 (0,0) [0,0] OVO (0,0] [0,0) Domain of fo : ] OO -OO X 5( fog) (x) = 11 + 2 No fraction can have a denominator of 0. So the domain of fog cannot include values of x such that 11 X -+2=0. In other words, if we solve 11 x -+2 =0 for X, we are finding an excluded value of X. Here is what we get when we solve x 14 + 2 =0 for x. 11 +2=0 11 . x = 0 . x x Multiplying both sides of the equation by X 11 .x+2 . x=0 .x Distributing x 11 + 2x = 0 Simplifying 2x = -11 X = - 11 2 So, we must exclude the value x = - 11 2 from the domain. And note that we must also exclude x = 0 since the denominator of - is 0 when x = 0. These are the only two values we must exclude from the domain. Therefore, the domain of fog is (-0. - _ )u (- 1. .) u co. co) . ANSWER 11 (fog) (x) =- 2x + 11 Domain of (fog) : (-09, - 14 ) u (-11. . ) u (0, 09).X Domain : x + 9 ( x ) Domain x = ( 0 ; 0 ) fog ( x ) X 4+ 2) fog ( * ) 2x + 1 set denominator to .O 2x . + 11 = 0 2X Domain 7 . 9 (x ) ( 0 , 0 ) Domain . fog ( x ) Final . answer :. CS Scanned with CamScanner