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)=x^2 - 6x + 7 $\frac{f(x + h)-f(x)}{h}=square$ (simplify your answer.)

Explanation:

Step1: Find f(x + h)

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

$$\begin{align*} f(x + h)&=(x + h)^2-6(x + h)+7\\ &=x^{2}+2xh+h^{2}-6x-6h + 7 \end{align*}$$

\]

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

\[

$$\begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-6x-6h + 7)-(x^{2}-6x + 7)\\ &=x^{2}+2xh+h^{2}-6x-6h + 7-x^{2}+6x - 7\\ &=2xh+h^{2}-6h \end{align*}$$

\]

Step3: Find the difference - quotient

\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-6h}{h}\\ &=\frac{h(2x + h-6)}{h}\\ &=2x+h - 6 \end{align*}$$

\]

Answer:

\(2x+h - 6\)