Question

given the following function, determine the difference quotient, $\frac{f(x + h)-f(x)}{h}$. $f(x)=-3x - 7$

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$:
$f(x + h)=-3(x + h)-7=-3x-3h - 7$

Step2: Calculate $f(x + h)-f(x)$

\[

$$\begin{align*} f(x + h)-f(x)&=(-3x-3h - 7)-(-3x - 7)\\ &=-3x-3h - 7 + 3x+7\\ &=-3h \end{align*}$$

\]

Step3: Compute the difference quotient

\[
\frac{f(x + h)-f(x)}{h}=\frac{-3h}{h}=-3
\]

Answer:

$-3$