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.