Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows. \[ \begin{array}{l} f(x)=x-5 \\ g(x)=3 x^{2} \end{array} \] Write the expressions for $(f \cdot g)(x)$ and $(f+g)(x)$ and evaluate $(f-g)(2)$. \[ \begin{array}{c} (f \cdot g)(x)= \\ (f+g)(x)= \\ (f-g)(2)= \end{array} \]

Step 1 ：Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows: $f(x)=x-5$ and $g(x)=3 x^{2}$.

Step 2 ：We are asked to find the expressions for $(f \cdot g)(x)$, $(f+g)(x)$, and $(f-g)(2)$.

Step 3 ：The expression $(f \cdot g)(x)$ is the product of the functions $f$ and $g$. This means we multiply the expressions for $f(x)$ and $g(x)$ together to get $3x^{3}-15x^{2}$.

Step 4 ：The expression $(f+g)(x)$ is the sum of the functions $f$ and $g$. This means we add the expressions for $f(x)$ and $g(x)$ together to get $3x^{2}+x-5$.

Step 5 ：The expression $(f-g)(2)$ is the difference of the functions $f$ and $g$ evaluated at $x=2$. This means we subtract the expression for $g(2)$ from the expression for $f(2)$ to get $-15$.

Step 6 ：\(\boxed{(f \cdot g)(x)= 3x^{3}-15x^{2}}\)

Step 7 ：\(\boxed{(f+g)(x)= 3x^{2}+x-5}\)

Step 8 ：\(\boxed{(f-g)(2)= -15}\)