Question

for the function ( f(x) = -6x + 9 ), evaluate and simplify the difference quotient. question help: ✉ message instructor ▶ next question

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 \( x\) with \( x + h \):
\( f(x + h)=-6(x + h)+9=-6x-6h + 9 \)

Step3: Substitute into the difference quotient

Substitute \( f(x + h) \) and \( f(x) \) into the 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:
\( (-6x-6h + 9)-(-6x + 9)=-6x-6h + 9 + 6x-9=-6h \)

Step5: Simplify the fraction

Now, divide the simplified numerator by \( h \):
\( \frac{-6h}{h}=-6 \) (since \( h
eq0 \), we can cancel \( h \))

Answer:

\( -6 \)