find the difference quotient of f; that is, find (f(x + h)-f(x))/h, h≠0, for the following function. f(x)=x²…

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x): [ \begin{align*} f(x + h)&=(x + h)^2-8(x + h)+9\ &=x^{2}+2xh+h^{2}-8x - 8h+9 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(x^{2}+2xh+h^{2}-8x - 8h+9)-(x^{2}-8x + 9)\ &=x^{2}+2xh+h^{2}-8x - 8h+9 - x^{2}+8x - 9\ &=2xh+h^{2}-8h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{2xh+h^{2}-8h}{h}\ &=\frac{h(2x + h-8)}{h}\ &=2x+h - 8 \end{align*} ]

Answer:

$2x+h - 8$