Find fog and gof if:
`f(x)=x^(2)+2,g(x)=1-1/(1-x),x!=1`

Explanation

To find \( f \circ g \) and \( g \circ f \) for the given functions \( f(x) = x^2 + 2 \) and \( g(x) = 1 - \frac{1}{1 - x} \), we will follow these steps:

### Step 1: Find \( f \circ g \)

The composition \( f \circ g \) means we need to find \( f(g(x)) \).

1. **Substitute \( g(x) \) into \( f(x) \)**:
\[ g(x) = 1 - \frac{1}{1 - x} \]
Now, substitute \( g(x) \) into \( f(x) \):
\[ f(g(x)) = f\left(1 - \frac{1}{1 - x}\right) \]
This means we need to calculate:
\[ f\left(1 - \frac{1}{1 - x}\right) = \left(1 - \frac{1}{1 - x}\right)^2 + 2 \]

2. **Simplify \( \left(1 - \frac{1}{1 - x}\right)^2 + 2 \)**:
First, simplify \( 1 - \frac{1}{1 - x} \):
\[ 1 - \frac{1}{1 - x} = \frac{(1 - x) - 1}{1 - x} = \frac{-x}{1 - x} \]
Now, square this:
\[ \left(\frac{-x}{1 - x}\right)^2 = \frac{x^2}{(1 - x)^2} \]
Now add 2:
\[ f(g(x)) = \frac{x^2}{(1 - x)^2} + 2 \]

3. **Combine into a single fraction**:
To combine \( \frac{x^2}{(1 - x)^2} + 2 \):
\[ = \frac{x^2 + 2(1 - x)^2}{(1 - x)^2} \]
Now expand \( 2(1 - x)^2 \):
\[ 2(1 - x)^2 = 2(1 - 2x + x^2) = 2 - 4x + 2x^2 \]
Therefore:
\[ f(g(x)) = \frac{x^2 + 2 - 4x + 2x^2}{(1 - x)^2} = \frac{3x^2 - 4x + 2}{(1 - x)^2} \]

### Step 2: Find \( g \circ f \)

Next, we need to find \( g(f(x)) \).

1. **Substitute \( f(x) \) into \( g(x) \)**:
\[ f(x) = x^2 + 2 \]
Now substitute \( f(x) \) into \( g(x) \):
\[ g(f(x)) = g(x^2 + 2) = 1 - \frac{1}{1 - (x^2 + 2)} \]
Simplifying:
\[ = 1 - \frac{1}{-x^2 - 1} = 1 + \frac{1}{x^2 + 1} \]

2. **Combine into a single fraction**:
\[ g(f(x)) = \frac{(x^2 + 1) + 1}{x^2 + 1} = \frac{x^2 + 2}{x^2 + 1} \]

### Final Results

Thus, we have:
- \( f \circ g = \frac{3x^2 - 4x + 2}{(1 - x)^2} \)
- \( g \circ f = \frac{x^2 + 2}{x^2 + 1} \)