Question

for the function ( f(x) = -6x + 9 ), evaluate and simplify the difference quotient.
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Explanation:

Step1: Recall the difference quotient formula

The difference quotient of a function \( f(x) \) is given by \( \frac{f(x + h)-f(x)}{h} \), where \( h
eq0 \).

Step2: Find \( f(x + h) \)

Given \( f(x)=-6x + 9 \), substitute \( x + h \) into the function:
\( f(x + h)=-6(x + h)+9=-6x-6h + 9 \)

Step3: Substitute into the difference quotient

Substitute \( f(x + h) \) and \( f(x) \) into the difference quotient formula:
\[

$$\begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{(-6x-6h + 9)-(-6x + 9)}{h}\\ \end{align*}$$

\]

Step4: Simplify the numerator

Simplify the numerator:
\[

$$\begin{align*} (-6x-6h + 9)-(-6x + 9)&=-6x-6h + 9 + 6x-9\\ &=-6h \end{align*}$$

\]

Step5: Simplify the difference quotient

Now divide the simplified numerator by \( h \):
\[
\frac{-6h}{h}=-6 \quad (h
eq0)
\]

Answer:

\( -6 \)