Given the following functions, find and simplify $$ \left(\frac{f}{g}\right)(-3) $$. $$ \begin{aligned} f(x)=x^{2}(3 x-2) \ g(x)=3 x-2\end{aligned} $$

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UpStudy AI · Accepted Answer

We are given the functions
$$
f(x)=x^{2}(3x-2) \quad \text{and} \quad g(x)=3x-2.
$$
Our goal is to compute
$$
\left(\frac{f}{g}\right)(-3)=\frac{f(-3)}{g(-3)},
$$
making sure that $$g(-3) \neq 0$$.

### Step 1: Evaluate $$ f(-3) $$

Substitute $$-3$$ into $$ f(x) $$:
$$
f(-3)=(-3)^{2} \cdot (3(-3)-2).
$$

Calculate $$ (-3)^2 $$:
$$
(-3)^2=9.
$$

Now calculate the second factor:
$$
3(-3) - 2 = -9 - 2 = -11.
$$

Thus,
$$
f(-3)=9 \cdot (-11)=-99.
$$

### Step 2: Evaluate $$ g(-3) $$

Substitute $$-3$$ into $$ g(x) $$:
$$
g(-3)=3(-3)-2.
$$

Calculate:
$$
3(-3)-2=-9-2=-11.
$$

### Step 3: Compute $$\left(\frac{f}{g}\right)(-3)$$

Now we have:
$$
\left(\frac{f}{g}\right)(-3)=\frac{-99}{-11}.
$$

Simplify the fraction:
$$
\frac{-99}{-11}=\frac{99}{11}=9.
$$

### Final Answer

$$
\left(\frac{f}{g}\right)(-3)=9.
$$
$$ \left(\frac{f}{g}\right)(-3) = 9 $$

Given the following functions, find and simplify \( \left(\frac{f}{g}\right)(-3) \). \[ \begin{aligned} f(x)=x^{2}(3 x-2) \\ g(x)=3 x-2\end{aligned} \]

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