Let's analyze the given graph and solve each part of the question step by step.

Part (a): Find $(g \circ f)(3)$

This means we need to evaluate $g(f(3))$.

1. Step 1: Find $f(3)$. From the graph, at $x = 3$, the value of $f(x)$ is $f(3) = 2$.

2. Step 2: Find $g(f(3)) = g(2)$. Now, look at the graph of $g(x)$. At $x = 2$, $g(2) = -1$.

Thus, $(g \circ f)(3) = -1$.

Part (b): Find $(f \circ g)(-3)$

This means we need to evaluate $f(g(-3))$.

1. Step 1: Find $g(-3)$. From the graph, at $x = -3$, the value of $g(x)$ is $g(-3) = 2$.

2. Step 2: Find $f(g(-3)) = f(2)$. Now, from the graph of $f(x)$, at $x = 2$, $f(2) = 3$.

Thus, $(f \circ g)(-3) = 3$.

Part (c): Find $(f \circ f)(-1)$

This means we need to evaluate $f(f(-1))$.

1. Step 1: Find $f(-1)$. From the graph of $f(x)$, at $x = -1$, $f(-1) = 2$.

2. Step 2: Find $f(f(-1)) = f(2)$. From the graph, we know that $f(2) = 3$.

Thus, $(f \circ f)(-1) = 3$.

Summary of Results:

• $(g \circ f)(3) = -1$
• $(f \circ g)(-3) = 3$
• $(f \circ f)(-1) = 3$

Would you like a more detailed explanation for any of these steps? Feel free to ask!

Here are 5 related questions to deepen your understanding:

1. How do we evaluate composite functions in general?
2. What is the significance of the domain and range in function composition?
3. How can you tell from a graph when a function is increasing or decreasing?
4. How would you handle composite functions if the functions involved were not graphed?
5. How does the behavior of $f$ and $g$ change as $x \to \infty$?

Tip: When evaluating composite functions, always begin by solving the inner function first before moving on to the outer function.