Find the difference quotient of \( f \), that is, find \( \frac{f(x+h)-f(x)}{h}, h \neq 0 \), for the following function. \( f(x)=-6 x+9 \) \( \frac{f(x+h)-f(x)}{h}=\square \) (Simplify your answer.)

To find the difference quotient \( \frac{f(x+h)-f(x)}{h} \) for the function \( f(x) = -6x + 9 \), we first calculate \( f(x + h) \). 1. **Calculate \( f(x + h) \)**: \[ f(x + h) = -6(x + h) + 9 = -6x - 6h + 9 \] 2. **Now, substitute this into the difference quotient**: \[ \frac{f(x+h) - f(x)}{h} = \frac{(-6x - 6h + 9) - (-6x + 9)}{h} \] 3. **Simplify the numerator**: \[ (-6x - 6h + 9) - (-6x + 9) = -6x - 6h + 9 + 6x - 9 = -6h \] 4. **Now, place this back into the quotient**: \[ \frac{-6h}{h} \] 5. **Since \( h \neq 0 \), we can simplify**: \[ -6 \] Thus, the simplified difference quotient is: \[ \frac{f(x+h)-f(x)}{h} = -6 \]