Suppose that the functions $$ f $$ and $$ g $$ are defined for all real numbers $$ x $$ as follows. $$ \begin{array}{l}f(x)=x^{3} \ g(x)=3 x^{2} \ (g \cdot f)(x)=\text { Write the expressions for }(g \cdot f)(x) \text { and }(g-f)(x) \text { and evaluate }(g+f)(2) \ (g-f)(x)=\square\end{array} $$ $$ (g+f)(2)=\square $$

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To solve the problem, we need to find the expressions for $$ (g \cdot f)(x) $$, $$ (g - f)(x) $$, and evaluate $$ (g + f)(2) $$.

1. **Finding $$ (g \cdot f)(x) $$**:
   The product of the functions $$ g $$ and $$ f $$ is given by:
   $$
   (g \cdot f)(x) = g(x) \cdot f(x)
   $$
   Substituting the definitions of $$ g(x) $$ and $$ f(x) $$:
   $$
   g(x) = 3x^2 \quad \text{and} \quad f(x) = x^3
   $$
   Therefore,
   $$
   (g \cdot f)(x) = (3x^2)(x^3) = 3x^{2+3} = 3x^5
   $$

2. **Finding $$ (g - f)(x) $$**:
   The difference of the functions $$ g $$ and $$ f $$ is given by:
   $$
   (g - f)(x) = g(x) - f(x)
   $$
   Substituting the definitions:
   $$
   (g - f)(x) = 3x^2 - x^3
   $$

3. **Finding $$ (g + f)(2) $$**:
   The sum of the functions $$ g $$ and $$ f $$ is given by:
   $$
   (g + f)(x) = g(x) + f(x)
   $$
   Substituting the definitions:
   $$
   (g + f)(x) = 3x^2 + x^3
   $$
   Now, we evaluate $$ (g + f)(2) $$:
   $$
   (g + f)(2) = 3(2^2) + (2^3) = 3(4) + 8 = 12 + 8 = 20
   $$

Now we can summarize the results:
- $$ (g \cdot f)(x) = 3x^5 $$
- $$ (g - f)(x) = 3x^2 - x^3 $$
- $$ (g + f)(2) = 20 $$

Thus, the final answers are:
$$
(g \cdot f)(x) = 3x^5, \quad (g - f)(x) = 3x^2 - x^3, \quad (g + f)(2) = 20
$$
$$ (g \cdot f)(x) = 3x^5 $$, $$ (g - f)(x) = 3x^2 - x^3 $$, and $$ (g + f)(2) = 20 $$.

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