Let's go through each part of the question step by step using the function $f(x) = 3x - 7$.

Part (A): $f(x + h)$

Substitute $x + h$ into the function:

$f(x + h) = 3(x + h) - 7$

Simplify:

$f(x + h) = 3x + 3h - 7$

So, the answer for Part (A) is:

$f(x + h) = 3x + 3h - 7$

Part (B): $f(x + h) - f(x)$

Using the expressions for $f(x + h)$ and $f(x)$:

$f(x + h) - f(x) = (3x + 3h - 7) - (3x - 7)$

Simplify:

$f(x + h) - f(x) = 3x + 3h - 7 - 3x + 7$

$f(x + h) - f(x) = 3h$

So, the answer for Part (B) is:

$f(x + h) - f(x) = 3h$

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

Using the result from Part (B):

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

Simplify:

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

So, the answer for Part (C) is:

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

Would you like to dive deeper into any of these calculations, or have any questions?

Follow-up Questions

1. How would the results change if the function were quadratic or more complex?
2. What do these calculations imply about the slope of the function $f(x) = 3x - 7$?
3. How can these steps be applied to finding the derivative of more complex functions?
4. How would the steps differ if the function included additional terms like $x^2$ or $\sin(x)$?
5. What is the significance of the limit of $\frac{f(x + h) - f(x)}{h}$ as $h \to 0$?

Tip: The expression $\frac{f(x + h) - f(x)}{h}$ as $h \to 0$ is fundamental to understanding the derivative of a function at a point.