find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=1/(x - 9) simplify…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{1}{(x + h)-9}=\frac{1}{x+h - 9}$

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

$f(x + h)-f(x)=\frac{1}{x + h-9}-\frac{1}{x - 9}=\frac{(x - 9)-(x + h - 9)}{(x + h-9)(x - 9)}=\frac{x-9-x - h + 9}{(x + h-9)(x - 9)}=\frac{-h}{(x + h-9)(x - 9)}$

Step3: Calculate the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{\frac{-h}{(x + h-9)(x - 9)}}{h}=\frac{-h}{(x + h-9)(x - 9)}\times\frac{1}{h}=-\frac{1}{(x + h-9)(x - 9)}$

Answer:

$-\frac{1}{(x + h-9)(x - 9)}$