find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=7/(x - 8) simplify…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: $f(x + h)=\frac{7}{(x + h)-8}=\frac{7}{x+h - 8}$

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

$f(x + h)-f(x)=\frac{7}{x + h-8}-\frac{7}{x - 8}=\frac{7(x - 8)-7(x + h - 8)}{(x + h-8)(x - 8)}=\frac{7x-56-(7x+7h - 56)}{(x + h-8)(x - 8)}=\frac{7x-56 - 7x-7h + 56}{(x + h-8)(x - 8)}=\frac{-7h}{(x + h-8)(x - 8)}$

Step3: Calculate the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{\frac{-7h}{(x + h-8)(x - 8)}}{h}=\frac{-7h}{h(x + h-8)(x - 8)}=-\frac{7}{(x + h-8)(x - 8)}$

Answer:

$-\frac{7}{(x + h-8)(x - 8)}$