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

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

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

Step 8 ：$\boxed{{\displaystyle (f-g)(2)=-15}}$