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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.

Find the difference quotient for the function f(x) = 4x + 7. Simplify your answer as much as possible.

Learn how to find the difference quotient for the linear function f(x) = 4x + 7 with a detailed step-by-step solution. This problem involves core concepts in algebra and calculus, suitable for students in Grades 10-12.

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