Find and simplify the following for

​f(x)equals=​x(2424minus−​x),

assuming

hnot equals≠0

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 EndFractionf(x+h)−f(x)h

Let's go through each part of the problem step by step. 

### Given function:
\[ f(x) = x(24 - x) \]

### (A) Find \( f(x + h) \)
We substitute \( x + h \) for \( x \) in the function \( f(x) \):
\[
f(x + h) = (x + h)(24 - (x + h))
\]

Now expand and simplify the expression:
\[
f(x + h) = (x + h)(24 - x - h) = x(24 - x - h) + h(24 - x - h)
\]
\[
f(x + h) = x(24 - x - h) + h(24 - x - h) = x(24 - x) - xh + h(24 - x) - h^2
\]
\[
f(x + h) = (24x - x^2 - xh) + (24h - hx - h^2)
\]
\[
f(x + h) = 24x - x^2 - xh + 24h - hx - h^2
\]
Since \( -xh \) and \( -hx \) are like terms, we combine them:
\[
f(x + h) = 24x - x^2 - 2xh + 24h - h^2
\]

So:
\[
f(x + h) = 24x - x^2 - 2xh + 24h - h^2
\]

### (B) Find \( f(x + h) - f(x) \)
Subtract \( f(x) \) from \( f(x + h) \):
\[
f(x + h) - f(x) = (24x - x^2 - 2xh + 24h - h^2) - (24x - x^2)
\]

Simplify by canceling out like terms:
\[
f(x + h) - f(x) = -2xh + 24h - h^2
\]

### (C) Find \( \frac{f(x + h) - f(x)}{h} \)
Divide the expression from part (B) by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = \frac{-2xh + 24h - h^2}{h}
\]

Simplify by dividing each term by \( h \):
\[
\frac{f(x + h) - f(x)}{h} = -2x + 24 - h
\]

Thus, the final answer is:
\[
\frac{f(x + h) - f(x)}{h} = -2x + 24 - h
\]

Would you like more details or have any questions?

Here are five related questions to further your understanding:

1. What happens to the expression \( \frac{f(x + h) - f(x)}{h} \) as \( h \) approaches 0?
2. How does the expression \( \frac{f(x + h) - f(x)}{h} \) relate to the derivative of \( f(x) \)?
3. Can you find the derivative of \( f(x) = x(24 - x) \) using the power rule?
4. What is the significance of the term \( h^2 \) in the expansion?
5. How would the problem change if \( f(x) \) were a different quadratic function?

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

Explore how to find and simplify the difference quotient for the function f(x) = x(24 - x), covering steps (A), (B), and (C). This problem involves quadratic functions and function transformations. Suitable for high school students.

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