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The question in the image appears to be about finding the function composition of two functions. Here's the content from the image:

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**Use function composition to determine \( f(g(x)) \) and \( g(f(x)) \) for the functions:**

\[ f(x) = 3x + 1 \quad \text{and} \quad g(x) = x^2 + 1 \]

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Let's compute both \( f(g(x)) \) and \( g(f(x)) \).

### 1. Finding \( f(g(x)) \):

Given that \( f(x) = 3x + 1 \) and \( g(x) = x^2 + 1 \), to find \( f(g(x)) \), we substitute \( g(x) \) into \( f(x) \):

\[
f(g(x)) = f(x^2 + 1)
\]
Now substitute \( x^2 + 1 \) into \( f(x) = 3x + 1 \):
\[
f(g(x)) = 3(x^2 + 1) + 1 = 3x^2 + 3 + 1 = 3x^2 + 4
\]

### 2. Finding \( g(f(x)) \):

To find \( g(f(x)) \), we substitute \( f(x) \) into \( g(x) \):
\[
g(f(x)) = g(3x + 1)
\]
Now substitute \( 3x + 1 \) into \( g(x) = x^2 + 1 \):
\[
g(f(x)) = (3x + 1)^2 + 1
\]
First, expand \( (3x + 1)^2 \):
\[
(3x + 1)^2 = 9x^2 + 6x + 1
\]
Now, add 1:
\[
g(f(x)) = 9x^2 + 6x + 1 + 1 = 9x^2 + 6x + 2
\]

### Final Results:
- \( f(g(x)) = 3x^2 + 4 \)
- \( g(f(x)) = 9x^2 + 6x + 2 \)

Would you like a deeper explanation or have any questions?

Here are some related questions:
1. What is the general concept of function composition in mathematics?
2. Can the composition of two functions be commutative, i.e., is \( f(g(x)) \) always equal to \( g(f(x)) \)?
3. How does the degree of a polynomial change after composing two functions?
4. How would you compute \( f(f(x)) \) or \( g(g(x)) \) for these functions?
5. What happens to the function composition when one of the functions is a linear function?

**Tip:** Always expand expressions carefully when performing function composition to avoid algebraic errors.

Use function composition to determine f(g(x)) and g(f(x)) for the functions: f(x) = 3x + 1 and g(x) = x^2 + 1.

Learn how to solve function composition problems with f(x) = 3x + 1 and g(x) = x^2 + 1. This problem covers key algebraic concepts, including how to compute f(g(x)) and g(f(x)). Suitable for high school students in Grades 9-12.

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