The problem requires evaluating composite functions using the graphs of two functions, $f(x)$ and $g(x)$. We need to find the values of $f(g(3))$, $g(f(1))$, $f(f(2))$, and $g(g(4))$ using the given graphs.

1. Problem Description

The problem requires evaluating composite functions using the graphs of two functions, $f(x)$ and $g(x)$. We need to find the values of $f(g(3))$, $g(f(1))$, $f(f(2))$, and $g(g(4))$ using the given graphs.

2. Solution Steps

(a) Find $f(g(3))$:

First, find $g(3)$ from the graph of $g(x)$. From the graph, $g(3) = 1$.

Next, find $f(1)$ from the graph of $f(x)$. From the graph, $f(1) = 5$.

Therefore, $f(g(3)) = f(1) = 5$.

(b) Find $g(f(1))$:

First, find $f(1)$ from the graph of $f(x)$. From the graph, $f(1) = 5$.

Next, find $g(5)$ from the graph of $g(x)$. From the graph, $g(5) = 3$.

Therefore, $g(f(1)) = g(5) = 3$.

(c) Find $f(f(2))$:

First, find $f(2)$ from the graph of $f(x)$. From the graph, $f(2) = 1$.

Next, find $f(1)$ from the graph of $f(x)$. From the graph, $f(1) = 5$.

Therefore, $f(f(2)) = f(1) = 5$.

(d) Find $g(g(4))$:

First, find $g(4)$ from the graph of $g(x)$. From the graph, $g(4) = 3$.

Next, find $g(3)$ from the graph of $g(x)$. From the graph, $g(3) = 1$.

Therefore, $g(g(4)) = g(3) = 1$.

3. Final Answer

$f(g(3)) = 5$

$g(f(1)) = 3$

$f(f(2)) = 5$

$g(g(4)) = 1$