Let's break down and solve the problem based on the given graph.

The problem involves:

• Two functions, $f(x)$ (red curve) and $g(x)$ (blue curve), whose graphs are shown.
• We are asked to find three composite function values:
  • (a) $(g \circ f)(-1)$
  • (b) $(f \circ g)(4)$
  • (c) $(f \circ f)(4)$

Step 1: Find the necessary function values from the graph.

(a) $(g \circ f)(-1)$

• This means we first find $f(-1)$, then use this result to find $g(f(-1))$.
  • From the graph, when $x = -1$, the value of $f(x)$ is $f(-1) = 3$.
  • Now, find $g(3)$ by checking the value of $g(x)$ when $x = 3$ on the blue graph.
  • From the graph, $g(3) = -3$.
  • Therefore, $(g \circ f)(-1) = -3$.

(b) $(f \circ g)(4)$

• This means we first find $g(4)$, then use this result to find $f(g(4))$.
  • From the graph, when $x = 4$, the value of $g(x)$ is $g(4) = 1$.
  • Now, find $f(1)$ by checking the value of $f(x)$ when $x = 1$ on the red graph.
  • From the graph, $f(1) = 3$.
  • Therefore, $(f \circ g)(4) = 3$.

(c) $(f \circ f)(4)$

• This means we first find $f(4)$, then use this result to find $f(f(4))$.
  • From the graph, when $x = 4$, the value of $f(x)$ is $f(4) = -2$.
  • Now, find $f(-2)$ by checking the value of $f(x)$ when $x = -2$ on the red graph.
  • From the graph, $f(-2) = 2$.
  • Therefore, $(f \circ f)(4) = 2$.

Final Answers:

• (a) $(g \circ f)(-1) = -3$
• (b) $(f \circ g)(4) = 3$
• (c) $(f \circ f)(4) = 2$

Would you like more details or have any other questions?

Here are 5 related questions for practice:

1. What is $(f \circ g)(0)$?
2. Find $g(g(2))$.
3. What is the value of $(g \circ f)(1)$?
4. If $h(x) = f(x) + g(x)$, what is $h(2)$?
5. Find $f(g(-2))$.

Tip: When working with composite functions, always evaluate the inner function first and then use its result to evaluate the outer function.