Suppose that the functions $f$ and $g$ are defined for all real numbers $x$ as follows.
\[
\begin{array}{l}
f(x)=x+6 \\
g(x)=4 x+2
\end{array}
\]
Write the expressions for $(g-f)(x)$ and $(g \cdot f)(x)$ and evaluate $(g+f)(4)$.
\[
\begin{array}{r}
(g-f)(x)=\square \\
(g \cdot f)(x)=\square \\
(g+f)(4)=\square
\end{array}
\]
Answer
For $(g+f)(4)$, we substitute $x = 4$ into $g(x)$ and $f(x)$, and then add the results together. We have $g(4) = 4 \cdot 4 + 2 = 18$ and $f(4) = 4 + 6 = 10$. Therefore, $(g+f)(4) = 18 + 10 = \boxed{28}$.
Step 1 ：We have $g(x) = 4x + 2$ and $f(x) = x + 6$.
Step 2 ：For $(g-f)(x)$, we subtract $f(x)$ from $g(x)$, which gives us $(4x + 2) - (x + 6) = 4x + 2 - x - 6 = 3x - 4$.
Step 3 ：For $(g \cdot f)(x)$, we multiply $g(x)$ by $f(x)$, which gives us $(4x + 2) \cdot (x + 6) = 4x^2 + 24x + 2x + 12 = 4x^2 + 26x + 12$.
Step 4 ：For $(g+f)(4)$, we substitute $x = 4$ into $g(x)$ and $f(x)$, and then add the results together. We have $g(4) = 4 \cdot 4 + 2 = 18$ and $f(4) = 4 + 6 = 10$. Therefore, $(g+f)(4) = 18 + 10 = \boxed{28}$.
