f(x) = x/(x+3)
g(x) = 2/x
Find the domain of the composite function f(g(x))

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!

Learn how to find the domain of the composite function f(g(x)) for f(x) = x/(x+3) and g(x) = 2/x. This problem covers key mathematical concepts like rational expressions, composite functions, and finding domains. Suitable for students in Grades 9-11.

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