Let's analyze the given problem step by step.

We are given the function: $f(x) = 8 + x + x^2$ and $h \neq 0$.

(a) Does $f(7 + 1) = f(7) + f(1)$?

We need to evaluate both sides.

• $f(7 + 1) = f(8)$

  • $f(8) = 8 + 8 + 8^2 = 8 + 8 + 64 = 80$

• Now, calculate $f(7) + f(1)$:

  • $f(7) = 8 + 7 + 7^2 = 8 + 7 + 49 = 64$
  • $f(1) = 8 + 1 + 1^2 = 8 + 1 + 1 = 10$
  • So, $f(7) + f(1) = 64 + 10 = 74$

Clearly, $f(8) \neq f(7) + f(1)$, so the answer is No.

(b) Find $f(x + h)$.

We need to substitute $x + h$ into the function $f(x) = 8 + x + x^2$: $f(x + h) = 8 + (x + h) + (x + h)^2$ Expanding the expression: $f(x + h) = 8 + x + h + (x^2 + 2xh + h^2)$ Simplifying: $f(x + h) = 8 + x + h + x^2 + 2xh + h^2$ Thus, $f(x + h) = 8 + x + x^2 + h + 2xh + h^2$.

(c) Does $f(x + h) = f(x) + f(h)$?

Let's calculate both sides.

• $f(x + h) = 8 + x + x^2 + h + 2xh + h^2$ (from part b).

• Now, calculate $f(x) + f(h)$:

  • $f(x) = 8 + x + x^2$
  • $f(h) = 8 + h + h^2$
  • So, $f(x) + f(h) = (8 + x + x^2) + (8 + h + h^2) = 16 + x + h + x^2 + h^2$

Clearly, $f(x + h) \neq f(x) + f(h)$ because the term $2xh$ is missing on the right-hand side. So, the answer is No.

(d) Does $f(x + h) = f(x) + h$?

Let's compare both sides.

• $f(x + h) = 8 + x + x^2 + h + 2xh + h^2$ (from part b).
• $f(x) + h = (8 + x + x^2) + h = 8 + x + x^2 + h$

Since $f(x + h)$ contains additional terms $2xh$ and $h^2$, the answer is No.

(e) Find $\frac{f(x + h) - f(x)}{h}$ and simplify.

First, let's compute the difference: $f(x + h) - f(x) = (8 + x + x^2 + h + 2xh + h^2) - (8 + x + x^2)$ Simplifying: $f(x + h) - f(x) = h + 2xh + h^2$ Now, divide by $h$: $\frac{f(x + h) - f(x)}{h} = \frac{h + 2xh + h^2}{h}$ Simplifying further: $\frac{f(x + h) - f(x)}{h} = 1 + 2x + h$

Thus, the simplified form is $1 + 2x + h$.

Do you need further clarifications or details on any step? Here are 5 related questions to explore:

1. How does the behavior of $f(x)$ change as $h \to 0$?
2. What is the derivative of $f(x) = 8 + x + x^2$?
3. Can $f(x)$ be factored in any useful way for solving future problems?
4. How does the quadratic term $x^2$ impact the difference quotient in part (e)?
5. How would you solve the limit of the difference quotient as $h \to 0$?

Tip: The difference quotient $\frac{f(x+h) - f(x)}{h}$ is a crucial step in finding the derivative of a function!