Find and simplify the following for ​f(x)equals​x(22minus​x), assuming hnot equals0 in ​(C).
​(A)​ f(xplus​h)
​(B)​ f(xplus​h)minus​f(x)
​(C) StartFraction f left parenthesis x plus h right parenthesis minus f left parenthesis x right parenthesis Over h EndFraction

Let's solve the given problem step by step.

Given the function \( f(x) = x(22 - x) \), we need to find and simplify the following:

### (A) \( f(x + h) \)
Substitute \( x + h \) into the function \( f(x) \):
\[
f(x + h) = (x + h)(22 - (x + h))
\]
Expand the expression:
\[
f(x + h) = (x + h)(22 - x - h)
\]
\[
f(x + h) = (x + h)(22 - x - h) = (x + h)(22 - x - h) = x(22 - x - h) + h(22 - x - h)
\]
\[
f(x + h) = x(22 - x - h) + h(22 - x - h) = x(22 - x - h) + h(22 - x - h) = 22x - x^2 - xh + 22h - xh - h^2
\]
\[
f(x + h) = 22x - x^2 - 2xh + 22h - h^2
\]

### (B) \( f(x + h) - f(x) \)
Subtract \( f(x) \) from \( f(x + h) \):
\[
f(x + h) - f(x) = (22x - x^2 - 2xh + 22h - h^2) - (22x - x^2)
\]
Simplify by canceling terms:
\[
f(x + h) - f(x) = -2xh + 22h - h^2
\]

### (C) \(\frac{f(x + h) - f(x)}{h}\)
Divide the result from (B) by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{-2xh + 22h - h^2}{h}
\]
Simplify by dividing each term by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = -2x + 22 - h
\]

### Summary:
- (A) \( f(x + h) = 22x - x^2 - 2xh + 22h - h^2 \)
- (B) \( f(x + h) - f(x) = -2xh + 22h - h^2 \)
- (C) \(\frac{f(x + h) - f(x)}{h} = -2x + 22 - h \)

Would you like further details on any step, or do you have any questions?

Here are five related questions for further exploration:
1. How would the results change if \( f(x) \) were a different quadratic function?
2. What is the significance of finding \( \frac{f(x + h) - f(x)}{h} \) in calculus?
3. How does the expression for \( f(x + h) \) relate to the concept of a derivative?
4. Can you determine the derivative of \( f(x) = x(22 - x) \) directly from your results?
5. What happens to \( \frac{f(x + h) - f(x)}{h} \) as \( h \) approaches 0?

**Tip:** The expression \( \frac{f(x + h) - f(x)}{h} \) is the difference quotient, a fundamental concept in finding the derivative of a function.

Explore how to simplify the function f(x) = x(22 - x) using the difference quotient method. Learn step-by-step solutions for finding f(x + h), f(x + h) - f(x), and (f(x + h) - f(x)) / h. Suitable for high school students studying calculus and algebra.

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