find the difference quotient of$$ \text{ } $$ $$ \frac{f\left(x+h\right)-f\left(x\right)}{h} $$ where $$ f\left(x\right)=-9x+3 $$ where h does not equal 0

To find the difference quotient of the function $ f(x) = -9x + 3 $, we need to compute the expression:

\[ \frac{f(x+h) - f(x)}{h} \]

where $ h \neq 0 $.

First, let's find $ f(x+h) $:

\[ f(x+h) = -9(x+h) + 3 \]
\[ f(x+h) = -9x - 9h + 3 \]

Now, we need to substitute $ f(x+h) $ and $ f(x) $ into the difference quotient:

\[ \frac{f(x+h) - f(x)}{h} = \frac{(-9x - 9h + 3) - (-9x + 3)}{h} \]

Simplify the numerator:

\[ = \frac{-9x - 9h + 3 + 9x - 3}{h} \]
\[ = \frac{-9h}{h} \]

Since $ h \neq 0 $, we can cancel $ h $ in the numerator and denominator:

\[ = -9 \]

Thus, the difference quotient is:

\[ \boxed{-9} \]

Would you like further details or have any questions? Here are some related questions for further exploration:

1. What is the definition of a difference quotient?
2. How does the difference quotient relate to the derivative of a function?
3. Compute the derivative of $ f(x) = -9x + 3 $ and verify it using the limit definition.
4. How would you find the difference quotient for a quadratic function, such as $ f(x) = x^2 $?
5. What is the geometric interpretation of the difference quotient?
6. Can the difference quotient be used for non-linear functions? Provide an example.
7. How does the difference quotient change if $ f(x) $ is a constant function?
8. What are some common applications of the difference quotient in calculus?

**Tip:** The difference quotient is a fundamental concept in calculus, forming the basis for the definition of the derivative. Understanding it deeply enhances your grasp of more advanced topics in calculus.

Learn how to calculate the difference quotient for the function f(x) = -9x + 3 step-by-step. This problem covers key calculus concepts, including the difference quotient formula. Suitable for high school students.

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