If $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$, find $f g$ and $( g \circ f ) ( x )$
A) $f g = x ^ { 3 } - 4 x ^ { 2 } + 9 x - 1 ; \quad \quad ( g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
B) $f g = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6 ; \quad (g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
C) $f g = 2 x ^ { 2 } + 3 x + 1$; $\quad $ $\quad $ $\quad $$( g \circ f ) ( x ) = 6 x ^ { 2 } + 7$
D) $f g = 6 x ^ { 2 } + 7$; $\quad $ $\quad $ $\quad $ $\quad $$( g \circ f ) ( x ) = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6$
E) none of these

Question

If $f ( x ) = 3 x - 2$ and $g ( x ) = 3 + 2 x ^ { 2 }$, find $f g$ and $( g \circ f ) ( x )$
A) $f g = x ^ { 3 } - 4 x ^ { 2 } + 9 x - 1 ; \quad \quad ( g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
B) $f g = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6 ; \quad  (g \circ f ) ( x ) = 18 x ^ { 2 } - 24 x + 11$
C) $f g = 2 x ^ { 2 } + 3 x + 1$; $\quad $ $\quad $ $\quad $$( g \circ f ) ( x ) = 6 x ^ { 2 } + 7$
D) $f g = 6 x ^ { 2 } + 7$; $\quad $ $\quad $ $\quad $ $\quad $$( g \circ f ) ( x ) = 6 x ^ { 3 } - 4 x ^ { 2 } + 9 x - 6$
E) none of these

Quizplus · Accepted Answer

To find $f g$, we substitute the definition of $g$ into $f$:
\begin{align*}
f g & = f ( g ( x ) ) \
& = f ( 3 + 2 x ^ { 2 } ) \
& = 3 ( 3 + 2 x ^ { 2 } ) - 2 \
& = 6 x ^ { 2 } + 9
\end{align*}
To find $( g \circ f ) ( x )$, we substitute the definition of $f$ into $g$:
\begin{align*}
( g \circ f ) ( x ) & = g ( f ( x ) ) \
& = g ( 3 x - 2 ) \
& = 3 + 2 ( 3 x - 2 ) ^ { 2 } \
& = 18 x ^ { 2 } - 24 x + 11
\end{align*}
Therefore, our answer is choice B.

If f(x)=3x−2f ( x ) = 3 x - 2f(x)=3x−2 And g(x)=3+2x2g ( x ) = 3 + 2 x ^ { 2 }g(x)=3+2x2

If $f ( x ) = 3 x - 2$ And $g ( x ) = 3 + 2 x ^ { 2 }$