Suppose that the functions $$ g $$ and $$ f $$ are defined as follows. $$ \begin{array}{l}g(x)=(x+3)(x+6) \ f(x)=3 x+8 \ \text { (a) Find }\left(\frac{g}{f}\right)(1) \text {. } \ \text { (b) Find all values that are NOT in the domain of } \frac{g}{f} \text {. } \ \text { If there is more than one value, separate them with commas. } \ \text { (a) }\left(\frac{g}{f}\right)(1)=\square \ \text { (b) Value(s) that are NOT in the domain of } \frac{g}{f}: \square\end{array} $$ a

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UpStudy AI · Accepted Answer

1. First, evaluate $$ g(1) $$ and $$ f(1) $$:
   $$
   g(1) = (1+3)(1+6) = 4 \times 7 = 28
   $$
   $$
   f(1) = 3(1) + 8 = 11
   $$
2. Then, compute $$\left(\frac{g}{f}\right)(1)$$:
   $$
   \left(\frac{g}{f}\right)(1) = \frac{g(1)}{f(1)} = \frac{28}{11}
   $$
3. The domain of $$\frac{g}{f}$$ includes all real numbers except those where $$ f(x) = 0 $$. Set $$ f(x) = 0 $$:
   $$
   3x + 8 = 0 \quad \Longrightarrow \quad 3x = -8 \quad \Longrightarrow \quad x = -\frac{8}{3}
   $$
4. Thus, the value not in the domain of $$\frac{g}{f}$$ is $$ -\frac{8}{3} $$.

$$
\text{(a) } \left(\frac{g}{f}\right)(1)=\frac{28}{11} \quad \text{(b) } -\frac{8}{3}
$$
$$ \text{(a) } \left(\frac{g}{f}\right)(1)=\frac{28}{11} \quad \text{(b) } -\frac{8}{3} $$

Suppose that the functions and are defined as follows.

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