Find and simplify each of the following for 

​f(x)equals=3 x minus 73x−7.

​(A) f left parenthesis x plus h right parenthesisf(x+h)

​(B) f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesisf(x+h)−f(x)

​(C) StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFractionf(x+h)−f(x)h

Question content area bottom

Part 1

​(A) 

f left parenthesis x plus h right parenthesisf(x+h)equals=3 x plus 3 h minus 73x+3h−7 

​(Do not​ factor.) 

Part 2

​(B) 

f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesisf(x+h)−f(x)equals=enter your response here 

​(Do not​ factor.) 

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.

Explore how to simplify expressions using function notation and compute the difference quotient. This problem focuses on applying function transformations and algebraic simplification techniques.

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