Use $$ f(x)=\frac{x}{x+1} $$ and $$ g(x)=3 x^{2}-2 $$ to find and simplify $$ (g \circ f)(x) $$ $$ (g \circ f)(x)=\frac{x^{2}-4 x-2}{x^{2}+2 x+1} $$ $$ (g \circ f)(x)=\frac{x-2}{x+1} $$ $$ (g \circ f)(x)=\frac{3 x^{2}-2}{3 x^{2}-1} $$ $$ (g \circ f)(x)=\frac{-4 x-2}{2 x+1} $$

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To find and simplify $$ (g \circ f)(x) $$, we need to compute $$ g(f(x)) $$.

1. **Identify the functions:**
   - $$ f(x) = \frac{x}{x+1} $$
   - $$ g(x) = 3x^2 - 2 $$

2. **Substitute $$ f(x) $$ into $$ g(x) $$:**
   $$
   g(f(x)) = g\left(\frac{x}{x+1}\right) = 3\left(\frac{x}{x+1}\right)^2 - 2
   $$

3. **Calculate $$ \left(\frac{x}{x+1}\right)^2 $$:**
   $$
   \left(\frac{x}{x+1}\right)^2 = \frac{x^2}{(x+1)^2} = \frac{x^2}{x^2 + 2x + 1}
   $$

4. **Substitute this back into $$ g(f(x)) $$:**
   $$
   g(f(x)) = 3\left(\frac{x^2}{x^2 + 2x + 1}\right) - 2
   $$

5. **Combine the terms:**
   $$
   g(f(x)) = \frac{3x^2}{x^2 + 2x + 1} - 2 = \frac{3x^2 - 2(x^2 + 2x + 1)}{x^2 + 2x + 1}
   $$

6. **Distribute the -2:**
   $$
   g(f(x)) = \frac{3x^2 - 2x^2 - 4x - 2}{x^2 + 2x + 1} = \frac{x^2 - 4x - 2}{x^2 + 2x + 1}
   $$

Now we have $$ (g \circ f)(x) = \frac{x^2 - 4x - 2}{x^2 + 2x + 1} $$.

Next, let's check if this can be simplified further.

7. **Factor the numerator and denominator if possible:**
   - The numerator $$ x^2 - 4x - 2 $$ does not factor nicely.
   - The denominator $$ x^2 + 2x + 1 = (x+1)^2 $$.

Thus, the simplified form of $$ (g \circ f)(x) $$ is:
$$
(g \circ f)(x) = \frac{x^2 - 4x - 2}{(x+1)^2}
$$

Now, let's check the provided options:
- $$ (g \circ f)(x) = \frac{x^2 - 4x - 2}{x^2 + 2x + 1} $$ is correct.
- The other options do not match this result.

Therefore, the final answer is:
$$
(g \circ f)(x) = \frac{x^2 - 4x - 2}{(x+1)^2}
$$
$$ (g \circ f)(x) = \frac{x^2 - 4x - 2}{(x+1)^2} $$

Use and to find and simplify

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