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Let's analyze the given problem. The functions provided are:

- \( f(x) = \frac{1 + x}{1 - x} \)
- \( g(x) = \frac{x}{1 - x} \)

You are asked to find \( f(g(x)) \) (the composition of \( f \) and \( g \)) and determine its domain.

### Step 1: Finding \( f(g(x)) \)

To find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \). That is:
\[
f(g(x)) = f\left( \frac{x}{1 - x} \right)
\]
We know \( f(x) = \frac{1 + x}{1 - x} \), so:
\[
f(g(x)) = \frac{1 + \frac{x}{1 - x}}{1 - \frac{x}{1 - x}}
\]
Now, simplify both the numerator and denominator.

#### Numerator:
\[
1 + \frac{x}{1 - x} = \frac{(1 - x) + x}{1 - x} = \frac{1}{1 - x}
\]

#### Denominator:
\[
1 - \frac{x}{1 - x} = \frac{(1 - x) - x}{1 - x} = \frac{1 - 2x}{1 - x}
\]

So, we now have:
\[
f(g(x)) = \frac{\frac{1}{1 - x}}{\frac{1 - 2x}{1 - x}} = \frac{1}{1 - 2x}
\]

Thus, \( f(g(x)) = \frac{1}{1 - 2x} \).

### Step 2: Finding the Domain of \( f(g(x)) \)

The function \( f(g(x)) = \frac{1}{1 - 2x} \) will be undefined when the denominator is zero, i.e., when \( 1 - 2x = 0 \). Solving for \( x \):

\[
1 - 2x = 0 \quad \Rightarrow \quad x = \frac{1}{2}
\]

Thus, the function \( f(g(x)) \) is defined for all real numbers except \( x = \frac{1}{2} \).

Additionally, since the function \( g(x) = \frac{x}{1 - x} \) has its own domain restriction (denominator \( 1 - x \neq 0 \)), we must also exclude \( x = 1 \) from the domain.

### Final Answer:
- \( f(g(x)) = \frac{1}{1 - 2x} \)
- The domain of \( f(g(x)) \) is \( x \in \mathbb{R} \setminus \left\{ 1, \frac{1}{2} \right\} \).

Would you like further details on any step?

Here are some related questions:
1. What is the composition of two arbitrary functions?
2. How do we find the domain of a composite function?
3. Why is division by zero undefined?
4. What are the conditions for a rational function to be defined?
5. How can we graph \( f(g(x)) \)?

**Tip**: Always check the domain restrictions of each individual function before composing them, as these restrictions will affect the final result.

f(x) = (1 + x)/(1 - x), g(x) = x/(1 - x). Find f(g(x)) and its domain.

Solve the problem of composing two rational functions, f(x) = (1 + x)/(1 - x) and g(x) = x/(1 - x), and find the domain of the resulting function. This problem covers function composition and domain restrictions, ideal for students in Grades 10-12.

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