To solve the given problem, we need to analyze the graphs of $f(x)$ and $g(x)$ based on the functions and perform the required calculations for compositions of functions.

Steps to Solve:

1. Find $(g \circ f)(-1)$: This means we first evaluate $f(-1)$ and then plug that result into $g(x)$.

   • From the graph, find $f(-1)$.
   • Once $f(-1)$ is known, evaluate $g(f(-1))$.

2. Find $(f \circ g)(1)$: This means we first evaluate $g(1)$ and then plug that result into $f(x)$.

   • From the graph, find $g(1)$.
   • Once $g(1)$ is known, evaluate $f(g(1))$.

3. Find $(f \circ f)(-1)$: This means we first evaluate $f(-1)$ and then plug that result into $f(x)$ again.

   • From the graph, find $f(-1)$.
   • Once $f(-1)$ is known, evaluate $f(f(-1))$.

Analysis Based on the Graph:

1. $f(-1)$: Looking at the red curve for $f(x)$, when $x = -1$, the value of $f(-1)$ is approximately $2$.

2. $g(2)$: Now, find $g(2)$ from the blue curve. When $x = 2$, the value of $g(2)$ appears to be around $-2$.

Thus, $(g \circ f)(-1) = g(f(-1)) = g(2) = -2$.

3. $g(1)$: Looking at the blue curve for $g(x)$, when $x = 1$, the value of $g(1)$ is $0$.

4. $f(0)$: Now, find $f(0)$ from the red curve. When $x = 0$, the value of $f(0)$ is also $0$.

Thus, $(f \circ g)(1) = f(g(1)) = f(0) = 0$.

5. $f(-1)$: As we saw earlier, $f(-1) = 2$.

6. $f(2)$: Now, find $f(2)$ from the red curve. When $x = 2$, the value of $f(2)$ is $3$.

Thus, $(f \circ f)(-1) = f(f(-1)) = f(2) = 3$.

Answers:

• (a) $(g \circ f)(-1) = -2$
• (b) $(f \circ g)(1) = 0$
• (c) $(f \circ f)(-1) = 3$

Would you like further details or explanations on any step?

Here are 5 related questions for practice:

1. What is $(g \circ g)(1)$?
2. Find $(f \circ f)(0)$.
3. What is $g(-2)$ based on the graph?
4. Calculate $(g \circ f)(2)$.
5. What is the value of $f(3)$?

Tip: When dealing with compositions, always start by evaluating the innermost function first!