https://file.aiquickdraw.com/mathbot/file/1725398355397rg10jwwj.jpg

The image contains a table with functions \( f(x) \) and \( g(x) \) defined for certain values of \( x \). The problem asks to find the values for the following:

- (a) \( g(f(1)) \)
- (b) \( f(g(4)) \)
- (c) \( f(f(3)) \)

### Step-by-Step Solution:

#### Given Table:

\[
\begin{array}{c|c|c|c}
x & f(x) & x & g(x) \\
\hline
1 & 4 & 1 & 2 \\
2 & 3 & 2 & 3 \\
3 & 1 & 3 & 4 \\
4 & 2 & 4 & 5 \\
\end{array}
\]

### (a) \( g(f(1)) \)

1. First, find \( f(1) \) from the table.
   \[
   f(1) = 4
   \]
2. Now, find \( g(4) \) using the result from the previous step.
   \[
   g(4) = 5
   \]
Thus, \( g(f(1)) = 5 \).

### (b) \( f(g(4)) \)

1. First, find \( g(4) \) from the table.
   \[
   g(4) = 5
   \]
2. Now, find \( f(5) \). However, the table does not provide a value for \( f(5) \), meaning \( f(5) \) is not defined based on the given data.

Thus, \( f(g(4)) \) is **undefined**.

### (c) \( f(f(3)) \)

1. First, find \( f(3) \) from the table.
   \[
   f(3) = 1
   \]
2. Now, find \( f(1) \) using the result from the previous step.
   \[
   f(1) = 4
   \]
Thus, \( f(f(3)) = 4 \).

### Summary:
- (a) \( g(f(1)) = 5 \)
- (b) \( f(g(4)) \) is **undefined**.
- (c) \( f(f(3)) = 4 \)

Would you like further details on this, or do you have any other questions?

### Related Questions:

1. What is the value of \( g(g(2)) \)?
2. What would \( f(f(1)) \) be?
3. How do you interpret \( g(g(x)) \) for a given \( x \)?
4. What if the table included \( f(5) = 6 \)? How would that change part (b)?
5. Can a function be defined such that both \( f(x) \) and \( g(x) \) are inverses of each other?

### Tip:
When dealing with nested functions like \( f(f(x)) \) or \( g(f(x)) \), always solve from the innermost function outward.

Given a table with functions f(x) and g(x) defined for certain values of x, find the values of: (a) g(f(1)), (b) f(g(4)), (c) f(f(3)).

Solve the function composition problems g(f(1)), f(g(4)), and f(f(3)) using the provided table of f(x) and g(x). This exercise is perfect for students in Grades 9-12 to understand the concepts of function evaluation and composition.

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