The problem asks us to compute $\left( \frac{f}{g} \right)(2)$, where:

$f(x) = x^3 - 2x - 3 \quad \text{and} \quad g(x) = 3x - 2.$

The function $\frac{f}{g}(x)$ is defined as:

$\frac{f}{g}(x) = \frac{f(x)}{g(x)}.$

So, to compute $\left( \frac{f}{g} \right)(2)$, we need to evaluate $f(2)$ and $g(2)$, then divide the results:

Step 1: Compute $f(2)$

$f(2) = 2^3 - 2(2) - 3 = 8 - 4 - 3 = 1.$

Step 2: Compute $g(2)$

$g(2) = 3(2) - 2 = 6 - 2 = 4.$

Step 3: Compute $\frac{f(2)}{g(2)}$

$\frac{f(2)}{g(2)} = \frac{1}{4}.$

Thus, $\left( \frac{f}{g} \right)(2) = \frac{1}{4}$.

Let me know if you need more details or have any other questions!

Here are 5 related questions for further practice:

1. What is $f'(x)$ for $f(x) = x^3 - 2x - 3$?
2. Find $\left( \frac{f}{g} \right)'(2)$ for the given functions.
3. Compute $\left( \frac{f}{g} \right)(-1)$ for the given functions.
4. If $h(x) = f(x) \times g(x)$, what is $h(2)$?
5. Determine the zeroes of $f(x) = x^3 - 2x - 3$.

Tip: Always evaluate functions step by step to avoid confusion with algebraic manipulations!