https://file.aiquickdraw.com/mathbot/file/1725553106810xckkncng.jpg

The problem asks to show that:

\[
\frac{f(x+h) - f(x)}{h} = 2x - 3 + h
\]

where the function \( f(x) = x^2 - 3x + 7 \).

### Step-by-Step Solution:

1. **Start by calculating \( f(x+h) \):**
   \[
   f(x+h) = (x+h)^2 - 3(x+h) + 7
   \]
   Expanding this expression:
   \[
   f(x+h) = x^2 + 2xh + h^2 - 3x - 3h + 7
   \]

2. **Now calculate \( f(x+h) - f(x) \):**
   Using \( f(x) = x^2 - 3x + 7 \), we subtract:
   \[
   f(x+h) - f(x) = \left( x^2 + 2xh + h^2 - 3x - 3h + 7 \right) - \left( x^2 - 3x + 7 \right)
   \]
   Simplifying this:
   \[
   f(x+h) - f(x) = 2xh + h^2 - 3h
   \]

3. **Divide by \( h \):**
   \[
   \frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 3h}{h}
   \]
   Simplify each term:
   \[
   \frac{f(x+h) - f(x)}{h} = 2x + h - 3
   \]

4. **Conclusion:**
   \[
   \frac{f(x+h) - f(x)}{h} = 2x - 3 + h
   \]
   This confirms the given statement.

Let me know if you need further explanation or details!

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Here are 5 related questions to deepen understanding:
1. What is the significance of finding \( \frac{f(x+h) - f(x)}{h} \) in calculus?
2. How does this relate to the concept of the derivative?
3. What is the derivative of the given function \( f(x) = x^2 - 3x + 7 \)?
4. What happens as \( h \to 0 \) in this expression?
5. How would you approach the problem if \( f(x) \) were a different type of function, like a cubic function?

**Tip:** The expression \( \frac{f(x+h) - f(x)}{h} \) is known as the difference quotient, which leads to the derivative of a function as \( h \to 0 \).

If f(x) = x^2 - 3x + 7, then show that (f(x+h) - f(x))/h = 2x - 3 + h.

Solve a problem using the difference quotient for the function f(x) = x^2 - 3x + 7. This algebraic concept introduces the difference quotient, a fundamental tool in calculus, showing that (f(x+h) - f(x))/h = 2x - 3 + h. Suitable for students in Grades 10-12.

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