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To solve the difference quotient for the given function \( f(x) = 3x + 4 \), we need to compute:

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

### Step-by-step solution:

1. **Find \( f(x + h) \):**
   Substitute \( x + h \) into the function \( f(x) = 3x + 4 \):
   \[
   f(x + h) = 3(x + h) + 4 = 3x + 3h + 4
   \]

2. **Compute the difference \( f(x + h) - f(x) \):**
   Now subtract \( f(x) \) from \( f(x + h) \):
   \[
   f(x + h) - f(x) = (3x + 3h + 4) - (3x + 4) = 3x + 3h + 4 - 3x - 4 = 3h
   \]

3. **Simplify the difference quotient:**
   Now divide by \( h \):
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{3h}{h} = 3
   \]

Thus, the simplified difference quotient is:

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

Would you like further clarification or more details on this process?

Here are 5 related questions to consider:
1. How does the difference quotient relate to the concept of a derivative?
2. Can we apply the difference quotient to non-linear functions, and how would the process differ?
3. What is the geometric interpretation of the difference quotient for a linear function like \( f(x) = 3x + 4 \)?
4. How does the constant term (like the \( +4 \) in this function) affect the difference quotient?
5. How can we use the difference quotient to approximate instantaneous rates of change?

**Tip:** The difference quotient is essentially the average rate of change over an interval, and it's a key step in finding the derivative of a function.

Find the difference quotient for the function f(x) = 3x + 4.

Learn how to compute the difference quotient for the linear function f(x) = 3x + 4. Follow a step-by-step process to simplify the expression and understand key algebraic concepts related to rates of change and linear functions. Ideal for high school algebra students.

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