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To solve this problem, we are asked to find the composition of two functions \( f \) and \( g \), given by:

\[
f(x) = \frac{2}{x+3} \quad \text{and} \quad g(x) = \frac{7}{x}
\]

We need to find \( (f \circ g)(x) \), which is the composition of \( f \) and \( g \). This means that we will substitute \( g(x) \) into \( f(x) \).

### Step-by-Step Solution:

1. **Substitute** \( g(x) = \frac{7}{x} \) into \( f(x) \):
   
   \[
   (f \circ g)(x) = f(g(x)) = f\left(\frac{7}{x}\right)
   \]

2. **Substitute** \( \frac{7}{x} \) for \( x \) in \( f(x) \):
   
   \[
   f\left(\frac{7}{x}\right) = \frac{2}{\frac{7}{x} + 3}
   \]

3. **Simplify the expression** inside the denominator:
   
   \[
   \frac{7}{x} + 3 = \frac{7}{x} + \frac{3x}{x} = \frac{7 + 3x}{x}
   \]

   So, the expression becomes:
   
   \[
   f\left(\frac{7}{x}\right) = \frac{2}{\frac{7 + 3x}{x}} = 2 \cdot \frac{x}{7 + 3x}
   \]

4. **Final simplified result**:
   
   \[
   (f \circ g)(x) = \frac{2x}{7 + 3x}
   \]

### Domain of \( f \circ g \):
To find the domain, we need to determine where the expression is defined. There are two cases to consider:

1. **From \( g(x) \)**: Since \( g(x) = \frac{7}{x} \), the function is undefined when \( x = 0 \) (division by zero).
   
2. **From \( f(g(x)) \)**: The expression \( \frac{2x}{7 + 3x} \) is undefined when the denominator \( 7 + 3x = 0 \), i.e., when \( x = -\frac{7}{3} \).

Thus, the domain is all real numbers except \( x = 0 \) and \( x = -\frac{7}{3} \).

### Domain in interval notation:
The domain is:
\[
(-\infty, -\frac{7}{3}) \cup (-\frac{7}{3}, 0) \cup (0, \infty)
\]

### Final Answer:
\[
(f \circ g)(x) = \frac{2x}{7 + 3x}
\]
\[
\text{Domain of } f \circ g: (-\infty, -\frac{7}{3}) \cup (-\frac{7}{3}, 0) \cup (0, \infty)
\]

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Let me know if you'd like further details or clarifications! Here are some related questions you might find interesting:

1. How do you determine the domain of a composed function in general?
2. Can you explain the difference between domain and range in function composition?
3. What happens if we reverse the composition and find \( (g \circ f)(x) \)?
4. How would the range of \( f \circ g \) be determined?
5. What role do asymptotes play in determining the behavior of the function \( f \circ g \)?

**Tip:** Always simplify the inner function first before composing with the outer function. This often helps in identifying any restrictions on the domain early on.

For the functions f(x) = 2/(x+3) and g(x) = 7/x, find the composition f ∘ g and simplify your answer as much as possible. Write the domain using interval notation.

Learn how to find and simplify the composition of two functions f(x) = 2/(x+3) and g(x) = 7/x. Understand how to determine the domain of the composed function in interval notation. Suitable for high school students in Grades 10-12.

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