Question

chapter 1 quiz (1.1,1.2, 1.4, 1.5, 1.7)
14 points possible answered: 8/14
question 9
let ( f(x) = 3x + 2 ) and ( g(x) = 3x^2 + 3x ).
after simplifying,
( (f circ g)(x) = )
( (g circ f)(x) = )
next question

Explanation:

Step1: Find \((f \circ g)(x)\)

The composition \((f \circ g)(x)\) means \(f(g(x))\). We substitute \(g(x)\) into \(f(x)\). Given \(f(x) = 3x + 2\) and \(g(x)=3x^{2}+3x\), we replace \(x\) in \(f(x)\) with \(3x^{2}+3x\).
\[

$$\begin{align*} f(g(x))&=3(3x^{2}+3x)+2\\ &= 9x^{2}+9x + 2 \end{align*}$$

\]

Step2: Find \((g \circ f)(x)\)

The composition \((g \circ f)(x)\) means \(g(f(x))\). We substitute \(f(x)\) into \(g(x)\). Given \(f(x) = 3x + 2\) and \(g(x)=3x^{2}+3x\), we replace \(x\) in \(g(x)\) with \(3x + 2\).
\[

$$\begin{align*} g(f(x))&=3(3x + 2)^{2}+3(3x + 2)\\ &=3(9x^{2}+12x + 4)+9x + 6\\ &=27x^{2}+36x+12 + 9x+6\\ &=27x^{2}+45x + 18 \end{align*}$$

\]

Answer:

\((f \circ g)(x)=\boldsymbol{9x^{2}+9x + 2}\)

\((g \circ f)(x)=\boldsymbol{27x^{2}+45x + 18}\)