We are given two functions $f(x)$ and $g(x)$ based on their graphs. Let’s solve each part step by step using the composition of functions.

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

This means we need to first find $f(3)$, and then substitute that value into $g(x)$.

1. From the graph, find $f(3)$. Looking at the graph of $f(x)$ (red curve), when $x = 3$, $f(3) = 0$.
2. Now, find $g(0)$. From the graph of $g(x)$ (blue line), when $x = 0$, $g(0) = -1$.

Thus, $(g \circ f)(3) = g(f(3)) = g(0) = -1$.

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

This means we need to first find $g(-3)$, and then substitute that into $f(x)$.

1. From the graph of $g(x)$ (blue line), when $x = -3$, $g(-3) = 0$.
2. Now, find $f(0)$. From the graph of $f(x)$ (red curve), when $x = 0$, $f(0) = 2$.

Thus, $(f \circ g)(-3) = f(g(-3)) = f(0) = 2$.

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

This means we need to first find $f(-1)$, and then substitute that into $f(x)$.

1. From the graph of $f(x)$ (red curve), when $x = -1$, $f(-1) = 2$.
2. Now, find $f(2)$. From the graph of $f(x)$, when $x = 2$, $f(2) = 3$.

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

Final answers:

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

Would you like further details or clarifications?

Here are 5 related questions:

1. How do you interpret a composite function like $(f \circ g)(x)$?
2. What is the graphical significance of function composition?
3. How does one identify function values directly from a graph?
4. How do vertical and horizontal transformations affect the graph of a function?
5. How can you determine the inverse of a function from a graph?

Tip: When solving composite functions, always evaluate the innermost function first before applying the outer function.