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The image asks to find the difference quotient for the function \( f(x) = 7x + 3 \), using the formula:

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

Let's go through the steps.

1. Start with the given function \( f(x) = 7x + 3 \).
2. Compute \( f(x + h) \):
   \[
   f(x + h) = 7(x + h) + 3 = 7x + 7h + 3
   \]
3. Now apply the difference quotient formula:
   \[
   \frac{f(x + h) - f(x)}{h} = \frac{(7x + 7h + 3) - (7x + 3)}{h}
   \]
4. Simplify the expression:
   \[
   \frac{(7x + 7h + 3) - 7x - 3}{h} = \frac{7h}{h}
   \]
5. Simplify further:
   \[
   \frac{7h}{h} = 7
   \]

So, the difference quotient simplifies to **7**.

Would you like more details or further explanation on any of the steps?

Here are five follow-up questions for deeper exploration:
1. What does the difference quotient represent in terms of the slope of a function?
2. How would the difference quotient change if the function were quadratic, e.g., \( f(x) = x^2 + 2x + 1 \)?
3. What is the geometric meaning of the result we obtained for the linear function?
4. How can the difference quotient be used to approximate the derivative?
5. What happens to the difference quotient as \( h \) approaches 0?

**Tip:** The difference quotient is a foundational concept in calculus, as it leads to the definition of the derivative!

Find the difference quotient for the function f(x) = 7x + 3.

Find the difference quotient of the linear function f(x) = 7x + 3. Follow the step-by-step solution and learn how the difference quotient leads to the concept of derivatives. Suitable for high school students studying calculus.

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