https://file.aiquickdraw.com/mathbot/file/17330881997370sm7xben.jpg

Let’s solve this problem step-by-step.

### Given:
1. \( f(x) = 3x - 1 \)
2. \( g(x) = x^2 + 2 \)

You are tasked to find:
- \( f(g(4)) \)
- \( g(f(4)) \)

---

### Step 1: Calculate \( g(4) \)
Using \( g(x) = x^2 + 2 \):
\[
g(4) = 4^2 + 2 = 16 + 2 = 18
\]

### Step 2: Calculate \( f(g(4)) \)
Now substitute \( g(4) = 18 \) into \( f(x) = 3x - 1 \):
\[
f(g(4)) = f(18) = 3(18) - 1 = 54 - 1 = 53
\]

So, \( f(g(4)) = 53 \).

---

### Step 3: Calculate \( f(4) \)
Using \( f(x) = 3x - 1 \):
\[
f(4) = 3(4) - 1 = 12 - 1 = 11
\]

### Step 4: Calculate \( g(f(4)) \)
Now substitute \( f(4) = 11 \) into \( g(x) = x^2 + 2 \):
\[
g(f(4)) = g(11) = 11^2 + 2 = 121 + 2 = 123
\]

So, \( g(f(4)) = 123 \).

---

### Final Answers:
- \( f(g(4)) = 53 \)
- \( g(f(4)) = 123 \)

Would you like more details or further explanation?

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#### Related Questions:
1. How would \( f(g(x)) \) change if \( g(x) = x + 5 \)?
2. What happens to \( g(f(x)) \) if \( f(x) = x^2 \)?
3. Can you generalize \( f(g(x)) \) for any \( x \)?
4. What is the domain of \( f(g(x)) \) and \( g(f(x)) \)?
5. How do the compositions behave if \( f(x) \) and \( g(x) \) are inverses?

#### Tip:
Always work inside-out when evaluating compositions of functions

Find f(g(4)) and g(f(4)) given f(x) = 3x - 1 and g(x) = x^2 + 2.

Step-by-step solution to find f(g(4)) and g(f(4)) using the given functions f(x) = 3x - 1 and g(x) = x^2 + 2. Understand function composition with detailed calculations suitable for Grades 9-12 students.

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