Let's analyze the question from the image step by step.

The problem gives two functions:

• $f(x) = 3x + 7$
• $g(x) = \frac{x^2}{2}$

We are asked to find the value of $f(g(4))$.

Step 1: Compute $g(4)$

Using the function for $g(x)$:

$g(4) = \frac{4^2}{2} = \frac{16}{2} = 8$

Step 2: Compute $f(g(4)) = f(8)$

Now substitute $g(4) = 8$ into the function $f(x)$:

$f(8) = 3(8) + 7 = 24 + 7 = 31$

Thus, the value of $f(g(4))$ is 31.

Would you like more details on this solution, or have any further questions?

Here are 5 related questions to expand this concept:

1. How would you solve for $f(g(5))$ using the same functions?
2. What would happen if the function $f(x)$ was modified to $2x + 5$?
3. How does changing the input for $g(x)$ affect the output of $f(g(x))$?
4. Can you think of any real-world applications for nested functions like this one?
5. What happens if we reverse the functions, finding $g(f(x))$?

Tip: When working with nested functions, always solve the inner function first and then substitute its result into the outer function.