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Let's solve the problem step by step.

We are given the functions:

\[
f(x) = 2 - x
\]
\[
g(x) = 3x^2 + x + 3
\]

We need to solve for the following:

### a. \((f \circ g)(x)\) 
This is the composition \(f(g(x))\), which means we substitute \(g(x)\) into \(f(x)\).

\[
f(g(x)) = 2 - g(x) = 2 - (3x^2 + x + 3)
\]
Simplifying:
\[
f(g(x)) = 2 - 3x^2 - x - 3 = -3x^2 - x - 1
\]

Thus, \((f \circ g)(x) = -3x^2 - x - 1\).

### b. \((g \circ f)(x)\)
This is the composition \(g(f(x))\), which means we substitute \(f(x)\) into \(g(x)\).

\[
g(f(x)) = 3(f(x))^2 + f(x) + 3
\]
Substituting \(f(x) = 2 - x\):
\[
g(f(x)) = 3(2 - x)^2 + (2 - x) + 3
\]
First, expand \((2 - x)^2\):
\[
(2 - x)^2 = 4 - 4x + x^2
\]
Now substitute into the equation:
\[
g(f(x)) = 3(4 - 4x + x^2) + (2 - x) + 3
\]
Distribute the 3:
\[
g(f(x)) = 12 - 12x + 3x^2 + 2 - x + 3
\]
Simplify:
\[
g(f(x)) = 3x^2 - 13x + 17
\]

Thus, \((g \circ f)(x) = 3x^2 - 13x + 17\).

### c. \((f \circ g)(2)\)
We already found \((f \circ g)(x)\) to be \(-3x^2 - x - 1\), so now we substitute \(x = 2\):

\[
f(g(2)) = -3(2)^2 - (2) - 1 = -3(4) - 2 - 1 = -12 - 2 - 1 = -15
\]

Thus, \((f \circ g)(2) = -15\).

### d. \((g \circ f)(2)\)
We already found \((g \circ f)(x)\) to be \(3x^2 - 13x + 17\), so now we substitute \(x = 2\):

\[
g(f(2)) = 3(2)^2 - 13(2) + 17 = 3(4) - 26 + 17 = 12 - 26 + 17 = 3
\]

Thus, \((g \circ f)(2) = 3\).

### Final Answers:
- a. \((f \circ g)(x) = -3x^2 - x - 1\)
- b. \((g \circ f)(x) = 3x^2 - 13x + 17\)
- c. \((f \circ g)(2) = -15\)
- d. \((g \circ f)(2) = 3\)

Would you like further explanations or have any questions?

Here are 5 related questions to expand your knowledge:
1. How do you interpret the domain of a composed function?
2. Can you find \((f \circ f)(x)\) and \((g \circ g)(x)\)?
3. What is the difference between function composition and regular multiplication of functions?
4. How does the order of composition affect the result of two functions?
5. How do you differentiate a composed function using the chain rule?

**Tip:** When composing functions, make sure to carefully substitute one function into the other, and always simplify the expression step by step to avoid errors.

For f(x) = 2 - x and g(x) = 3x^2 + x + 3, find the following functions: a. (f ∘ g)(x), b. (g ∘ f)(x), c. (f ∘ g)(2), d. (g ∘ f)(2)

Learn how to solve function composition problems with f(x) = 2 - x and g(x) = 3x^2 + x + 3. Detailed solutions for a. (f ∘ g)(x), b. (g ∘ f)(x), c. (f ∘ g)(2), and d. (g ∘ f)(2), using polynomial and linear functions. Suitable for high school students in grades 9-12.

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