The question asks to simplify expressions involving the function $f(x) = 3x^2 - x + 2$. Let's solve the problem step by step.

Given:

$f(x) = 3x^2 - x + 2$

(a) Find $f(a+1)$:

To compute $f(a+1)$, substitute $x = a+1$ into the function:

$f(a+1) = 3(a+1)^2 - (a+1) + 2$

Now expand the expression:

$f(a+1) = 3(a^2 + 2a + 1) - (a + 1) + 2$ $f(a+1) = 3a^2 + 6a + 3 - a - 1 + 2$ $f(a+1) = 3a^2 + 5a + 4$

(b) Find $f(a) + f(1)$:

First, calculate $f(a)$ and $f(1)$ individually:

• $f(a) = 3a^2 - a + 2$
• $f(1) = 3(1)^2 - 1 + 2 = 3 - 1 + 2 = 4$

Now sum these:

$f(a) + f(1) = (3a^2 - a + 2) + 4 = 3a^2 - a + 6$

(c) Find $f(1+h) - f(1)$:

First, calculate $f(1+h)$ by substituting $x = 1+h$ into the function:

$f(1+h) = 3(1+h)^2 - (1+h) + 2$ $f(1+h) = 3(1 + 2h + h^2) - (1 + h) + 2$ $f(1+h) = 3 + 6h + 3h^2 - 1 - h + 2$ $f(1+h) = 3h^2 + 5h + 4$

Now, subtract $f(1) = 4$:

$f(1+h) - f(1) = (3h^2 + 5h + 4) - 4 = 3h^2 + 5h$

Do you need further details on any step, or have additional questions? Here are 5 related questions to consider:

1. How do you expand binomials like $(a+1)^2$?
2. What are the steps for simplifying polynomial expressions?
3. Can you describe the process for calculating $f(x)$ for specific values of $x$?
4. How is the difference $f(1+h) - f(1)$ related to the concept of derivatives?
5. What happens when the function is evaluated at negative numbers?

Tip: When simplifying polynomial expressions, always carefully expand and combine like terms to avoid mistakes.