Questions: Find the difference quotient (f(x+h)-f(x))/h, where h ≠ 0, for the function below. f(x)=1/(x-8) Simplify your answer as much as possible. (f(x+h)-f(x))/h=

Solution Steps

The given function is \( f(x) = \frac{1}{x-8} \).

The difference quotient formula is: \[ \frac{f(x+h) - f(x)}{h} \]

First, find \( f(x+h) \): \[ f(x+h) = \frac{1}{(x+h)-8} = \frac{1}{x+h-8} \]

Now, substitute \( f(x) \) and \( f(x+h) \) into the difference quotient formula: \[ \frac{\frac{1}{x+h-8} - \frac{1}{x-8}}{h} \]

Combine the fractions in the numerator: \[ \frac{\frac{1}{x+h-8} - \frac{1}{x-8}}{h} = \frac{\frac{(x-8) - (x+h-8)}{(x+h-8)(x-8)}}{h} \]

Simplify the numerator: \[ \frac{\frac{x-8 - x - h + 8}{(x+h-8)(x-8)}}{h} = \frac{\frac{-h}{(x+h-8)(x-8)}}{h} \]

Cancel out \( h \) in the numerator and denominator: \[ \frac{-h}{h(x+h-8)(x-8)} = \frac{-1}{(x+h-8)(x-8)} \]

Final Answer