Question

find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h ≠ 0, for the following function. be sure to simplify.
f(x)=x² - 9x + 5
$\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x):
\[

$$\begin{align*} f(x + h)&=(x + h)^2-9(x + h)+5\\ &=x^{2}+2xh+h^{2}-9x - 9h+5 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-9x - 9h+5)-(x^{2}-9x + 5)\\ &=x^{2}+2xh+h^{2}-9x - 9h+5 - x^{2}+9x - 5\\ &=2xh+h^{2}-9h \end{align*}$$

\]

Step3: Find the difference quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-9h}{h}\\ &=\frac{h(2x + h-9)}{h}\\ &=2x+h - 9 \end{align*}$$

\]

Answer:

$2x+h - 9$