We are given a table with values of $f(x)$ and $g(x)$ for different values of $x$. The task is to evaluate $f(g(1))$ and $g(f(4))$.

Let's break this down:

1. To find $f(g(1))$:

• First, we need to find $g(1)$. From the table, when $x = 1$, $g(1) = 2$.
• Now, substitute this into $f(g(1))$, which becomes $f(2)$. From the table, $f(2) = 9$.

So, $f(g(1)) = 9$.

2. To find $g(f(4))$:

• First, we need to find $f(4)$. From the table, when $x = 4$, $f(4) = 3$.
• Now, substitute this into $g(f(4))$, which becomes $g(3)$. From the table, $g(3) = 3$.

So, $g(f(4)) = 3$.

Final answers:

• $f(g(1)) = 9$
• $g(f(4)) = 3$

Would you like further explanation or details?

Here are 5 related questions to expand on this:

1. How can we use function composition in solving real-world problems?
2. Can $f$ and $g$ be the same function? What would happen in that case?
3. How do we define the domain and range of composite functions?
4. What would happen if one of the values was not defined in the table?
5. How can you generalize the process of evaluating composite functions for any functions $f$ and $g$?

Tip: Always check intermediate function values carefully to avoid errors when solving composite functions.