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.