To find the domain of the composite function $f(g(x))$, we need to follow these steps:

1. Find the domain of $g(x)$: The function $g(x) = \frac{2}{x}$ is defined for all $x$ except where the denominator is zero. So, the domain of $g(x)$ is: $x \neq 0$

2. Find the composite function $f(g(x))$: We substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(\frac{2}{x}\right) = \frac{\frac{2}{x}}{\frac{2}{x} + 3}$ Simplifying this expression: $f(g(x)) = \frac{\frac{2}{x}}{\frac{2 + 3x}{x}} = \frac{2}{2 + 3x}$

3. Find the domain of $f(g(x))$: Now, we need to determine where this expression is undefined. The denominator of $f(g(x))$ is $2 + 3x$, and this is undefined when: $2 + 3x = 0 \quad \Rightarrow \quad x = -\frac{2}{3}$ Therefore, $x \neq -\frac{2}{3}$.

4. Combine the restrictions: The domain of $f(g(x))$ will be the values of $x$ where both $g(x)$ and $f(g(x))$ are defined. From step 1, $x \neq 0$, and from step 3, $x \neq -\frac{2}{3}$.

Thus, the domain of $f(g(x))$ is: $x \in \mathbb{R} \setminus \left\{0, -\frac{2}{3}\right\}$ Or in interval notation: $(-\infty, -\frac{2}{3}) \cup (-\frac{2}{3}, 0) \cup (0, \infty)$

Would you like more details or have any questions? Here are some related questions for further practice:

1. How would the domain change if $f(x) = \frac{x+1}{x+3}$?
2. Find the range of $f(g(x))$ for $f(x) = \frac{x}{x+3}$ and $g(x) = \frac{2}{x}$.
3. Determine the domain of $f(h(x))$ if $h(x) = \sqrt{x}$ and $f(x) = \frac{x}{x+3}$.
4. If $g(x) = x - 2$, what is the domain of $f(g(x))$ when $f(x) = \frac{x}{x+3}$?
5. Solve for the composite function $f(g(x))$ if $f(x) = \frac{x}{x+2}$ and $g(x) = \frac{1}{x}$.

Tip: Always analyze the domain restrictions of individual functions before forming composite functions to avoid undefined values!