To solve this problem, we need to find the difference quotient for the given function $f(x) = 4x + 7$. The difference quotient is defined as:

$\frac{f(x+h) - f(x)}{h}$

Step-by-step solution:

1. Find $f(x+h)$: Using the function $f(x) = 4x + 7$, replace $x$ with $x+h$:

   $f(x+h) = 4(x+h) + 7 = 4x + 4h + 7$

2. Subtract $f(x)$ from $f(x+h)$:

   $f(x+h) - f(x) = (4x + 4h + 7) - (4x + 7)$ Simplifying this:

   $f(x+h) - f(x) = 4x + 4h + 7 - 4x - 7 = 4h$

3. Divide by $h$:

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

Final Answer:

The simplified difference quotient is $\boxed{4}$.

Would you like more details on this, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. What is the geometric meaning of the difference quotient in relation to the slope of a line?
2. How does the difference quotient change if the function is quadratic, for example $f(x) = x^2$?
3. Why does the constant term $7$ in the function $f(x) = 4x + 7$ not affect the difference quotient?
4. How can the difference quotient be used to approximate the derivative of a function?
5. What happens to the difference quotient as $h$ approaches 0?

Tip: The difference quotient is a foundational concept in calculus because it leads to the derivative, which represents the instantaneous rate of change or the slope of the tangent line at a point.