The **difference quotient** is defined as:

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

For the given function \( f(x) = -3x - x \), simplify it first:

\[
f(x) = -3x - x = -4x
\]

Now, applying the difference quotient formula:

1. **Find \( f(x+h) \):**

\[
f(x+h) = -4(x+h) = -4x - 4h
\]

2. **Set up the difference quotient:**

\[
\frac{f(x+h) - f(x)}{h} = \frac{(-4x - 4h) - (-4x)}{h}
\]

3. **Simplify:**

\[
= \frac{-4x - 4h + 4x}{h} = \frac{-4h}{h}
\]

4. **Final simplification:**

\[
= -4
\]

So, the difference quotient for \( f(x) = -4x \) is \( -4 \).

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the difference quotient relate to the derivative?
2. What does the constant result of the difference quotient indicate about the function?
3. Can you calculate the derivative of \( f(x) = -4x \) directly?
4. How would the difference quotient change for a quadratic function?
5. How does the difference quotient help in understanding slope?

**Tip**: The difference quotient gives the average rate of change, and as \( h \) approaches zero, it leads to the derivative, showing the instantaneous rate of change.

Explore how to apply the difference quotient to the linear function f(x) = -3x - x. This step-by-step guide simplifies the function, walks through key algebraic concepts like rate of change, and explains how the difference quotient leads to the derivative. Ideal for students in Grades 10-12.

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