Find and simplify.
f(x) = 7 − x2
(a)
f(x + h) − f(x)

(b)
f(x + h) − f(x)
h

Let's work through both parts of the problem, step by step.

### Given: 
\( f(x) = 7 - x^2 \)

### (a) \( f(x + h) - f(x) \)

1. **Find \( f(x + h) \):**  
   Replace \( x \) with \( x + h \) in the function \( f(x) \).
   \[
   f(x + h) = 7 - (x + h)^2
   \]
   Expanding the square:
   \[
   f(x + h) = 7 - (x^2 + 2xh + h^2) = 7 - x^2 - 2xh - h^2
   \]

2. **Subtract \( f(x) \) from \( f(x + h) \):**
   \[
   f(x + h) - f(x) = \left( 7 - x^2 - 2xh - h^2 \right) - (7 - x^2)
   \]
   Simplifying:
   \[
   f(x + h) - f(x) = 7 - x^2 - 2xh - h^2 - 7 + x^2
   \]
   Cancel out terms:
   \[
   f(x + h) - f(x) = -2xh - h^2
   \]
   
Thus, the simplified expression for \( f(x + h) - f(x) \) is:
\[
f(x + h) - f(x) = -2xh - h^2
\]

### (b) \( \frac{f(x + h) - f(x)}{h} \)

Now, take the result from part (a) and divide by \( h \):

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

Simplify by dividing each term by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = -2x - h
\]

Thus, the simplified expression for \( \frac{f(x + h) - f(x)}{h} \) is:
\[
\frac{f(x + h) - f(x)}{h} = -2x - h
\]

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Would you like further details or have any questions? Here are some follow-up questions for practice:

1. What would \( f(x + h) - f(x) \) look like for a different quadratic function?
2. How would the expression change if the original function were linear?
3. What happens to the result in part (b) as \( h \to 0 \)?
4. Can you find the derivative of \( f(x) = 7 - x^2 \) using part (b)?
5. How does the process change if \( f(x) \) is cubic rather than quadratic?

**Tip:** When simplifying expressions, always look for common terms to cancel out before dividing.

Find and simplify.
f(x) = 7 − x^2
(a) f(x + h) − f(x)
(b) f(x + h) − f(x) / h

Learn how to simplify the difference quotient for the quadratic function f(x) = 7 − x^2. This example covers key algebraic concepts, including expanding and simplifying quadratic expressions, and introduces the limit definition of a derivative. Suitable for students in Grades 10-12.

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