find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h≠0, for the following function…

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)