The problem asks about the composition of two functions, $f(x)$ and $g(x)$, whose graphs are given in the image. Let's break down how to solve the two parts of the question.

The composition of functions, such as $(g \circ f)(x)$, means that you first apply the function $f(x)$, and then apply the function $g(x)$ to the result.

1. Part (a): $(g \circ f)(x)$

This is calculated as:

$(g \circ f)(x) = g(f(x))$

You first find $f(x)$ from the graph of $f$, and then use that value to find $g(f(x))$ from the graph of $g$.

Let's solve it step-by-step for different values of $x$:

$x = 1$:

• From the graph of $f(x)$, $f(1) = 1$.
• Now, use this value in the graph of $g$. $g(1) = 3$.
• Therefore, $(g \circ f)(1) = g(f(1)) = 3$.

$x = 2$:

• From the graph of $f(x)$, $f(2) = 2$.
• Now, $g(2) = 4$.
• Therefore, $(g \circ f)(2) = g(f(2)) = 4$.

$x = 3$:

• From the graph of $f(x)$, $f(3) = 1$.
• Now, $g(1) = 3$ (as before).
• Therefore, $(g \circ f)(3) = g(f(3)) = 3$.

$x = 4$:

• From the graph of $f(x)$, $f(4) = 4$.
• Now, $g(4) = 2$.
• Therefore, $(g \circ f)(4) = g(f(4)) = 2$.

So the values for $(g \circ f)(x)$ are: $(g \circ f)(1) = 3, \quad (g \circ f)(2) = 4, \quad (g \circ f)(3) = 3, \quad (g \circ f)(4) = 2.$

2. Part (b): $(f \circ g)(x)$

This is calculated as:

$(f \circ g)(x) = f(g(x))$

Now, we will use $g(x)$ first and then apply $f(x)$.

$x = 1$:

• From the graph of $g(x)$, $g(1) = 3$.
• Now, from the graph of $f(x)$, $f(3) = 1$.
• Therefore, $(f \circ g)(1) = f(g(1)) = 1$.

$x = 2$:

• From the graph of $g(x)$, $g(2) = 4$.
• Now, $f(4) = 4$.
• Therefore, $(f \circ g)(2) = f(g(2)) = 4$.

$x = 3$:

• From the graph of $g(x)$, $g(3) = 2$.
• Now, $f(2) = 2$.
• Therefore, $(f \circ g)(3) = f(g(3)) = 2$.

$x = 4$:

• From the graph of $g(x)$, $g(4) = 1$.
• Now, $f(1) = 1$.
• Therefore, $(f \circ g)(4) = f(g(4)) = 1$.

So the values for $(f \circ g)(x)$ are: $(f \circ g)(1) = 1, \quad (f \circ g)(2) = 4, \quad (f \circ g)(3) = 2, \quad (f \circ g)(4) = 1.$

Would you like more details on any of the steps? Here are some related questions for further practice:

1. What is the value of $(f \circ f)(x)$ for different values of $x$?
2. What is the value of $(g \circ g)(x)$?
3. How would you find the inverse of $f(x)$ and $g(x)$ based on the graphs?
4. Can you generalize a formula for $(g \circ f)(x)$ if $f(x)$ and $g(x)$ are given as functions?
5. What is the domain and range of $(g \circ f)(x)$ and $(f \circ g)(x)$?

Tip: When working with function compositions, it's helpful to break down each step and focus on applying one function at a time.