Let's break down the problem and solve both parts step-by-step:

Given the function $f(x) = 2 - x^2$, we need to evaluate:

(a) $f(x + 1)$

This means we need to replace $x$ with $x + 1$ in the expression for $f(x)$:

$f(x + 1) = 2 - (x + 1)^2$

Now, expand $(x + 1)^2$:

$f(x + 1) = 2 - (x^2 + 2x + 1)$

Simplify the expression:

$f(x + 1) = 2 - x^2 - 2x - 1$

$f(x + 1) = 1 - x^2 - 2x$

Thus, the correct expression for $f(x + 1)$ is:

$f(x + 1) = 1 - x^2 - 2x$

(b) $f(x) + f(1)$

We are asked to find $f(x) + f(1)$. First, evaluate $f(1)$:

$f(1) = 2 - 1^2 = 2 - 1 = 1$

Now, add $f(x)$ and $f(1)$:

$f(x) + f(1) = (2 - x^2) + 1$

Simplify:

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

Thus, the correct expression for $f(x) + f(1)$ is:

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

---

Summary of Answers:

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

Would you like further clarification or details on any step? Here are some additional related questions:

1. How would you evaluate $f(x - 1)$?
2. What would happen if the function was $f(x) = 2 + x^2$ instead?
3. How do you graph the function $f(x) = 2 - x^2$?
4. What are the roots of the equation $f(x) = 0$?
5. How does the function $f(x) = 2 - x^2$ behave as $x$ approaches infinity?

Tip: Always remember to carefully handle parentheses when substituting expressions like $(x + 1)$, as it affects both the coefficient of $x$ and the constant term when expanded.