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Let's analyze and solve the given functions and their compositions.

We are provided with:

\[
f(x) = 3x + 1
\]
\[
g(x) = \frac{x + 4}{2}
\]

The tasks are:

1. \( f \circ g(x) \) (i.e., \( f(g(x)) \))
2. \( g \circ f(x) \) (i.e., \( g(f(x)) \))

### 1. \( f \circ g(x) \)

We need to substitute \( g(x) \) into \( f(x) \).

\[
f(g(x)) = f\left( \frac{x + 4}{2} \right)
\]

Now, substitute \( \frac{x + 4}{2} \) into \( f(x) = 3x + 1 \):

\[
f\left( \frac{x + 4}{2} \right) = 3\left( \frac{x + 4}{2} \right) + 1
\]

Simplify:

\[
= \frac{3(x + 4)}{2} + 1 = \frac{3x + 12}{2} + 1
\]
\[
= \frac{3x + 12}{2} + \frac{2}{2} = \frac{3x + 14}{2}
\]

Thus, \( f \circ g(x) = \frac{3x + 14}{2} \).

### 2. \( g \circ f(x) \)

Now, substitute \( f(x) \) into \( g(x) \).

\[
g(f(x)) = g(3x + 1)
\]

Substitute \( 3x + 1 \) into \( g(x) = \frac{x + 4}{2} \):

\[
g(3x + 1) = \frac{(3x + 1) + 4}{2}
\]

Simplify:

\[
= \frac{3x + 1 + 4}{2} = \frac{3x + 5}{2}
\]

Thus, \( g \circ f(x) = \frac{3x + 5}{2} \).

### Final Answers:

1. \( f \circ g(x) = \frac{3x + 14}{2} \)
2. \( g \circ f(x) = \frac{3x + 5}{2} \)

Would you like further details or clarifications?

Here are five related questions:

1. What is the general method to find the composition of two functions?
2. How does composition of functions differ from simple addition or multiplication of functions?
3. What are some real-world applications of function composition?
4. How do you find the domain of a composed function?
5. How can inverse functions relate to function composition?

**Tip:** Always check the domain of the individual functions before composing them to ensure the composition is valid for all \(x\).

Given two functions f(x) = 3x + 1 and g(x) = (x + 4) / 2, calculate the compositions f(g(x)) and g(f(x)).

Solve the function composition problem where f(x) = 3x + 1 and g(x) = (x + 4) / 2. Learn how to find f(g(x)) and g(f(x)) step-by-step, using algebraic methods. Suitable for students in Grades 10-12.

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