To solve the given problems, we first need to understand the composition of functions and use the provided graphs of $y = f(x)$ (blue line) and $y = g(x)$ (red line) to determine the values.

Step-by-Step Solution:

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

1. Find $g(4)$: Locate $x = 4$ on the x-axis and find the corresponding y-value on the red line, which is the graph of $g(x)$.
2. Find $f(g(4))$: After finding $g(4)$, use this y-value as the new x-value to locate $f(g(4))$ on the blue line, which is the graph of $f(x)$.

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

1. Find $f(3)$: Locate $x = 3$ on the x-axis and find the corresponding y-value on the blue line, which represents $f(x)$.
2. Find $g(f(3))$: Use the result of $f(3)$ as the new x-value to locate $g(f(3))$ on the red line, which represents $g(x)$.

(c) $(f \circ f)(2)$

1. Find $f(2)$: Locate $x = 2$ on the x-axis and find the corresponding y-value on the blue line, which is the graph of $f(x)$.
2. Find $f(f(2))$: Use the result of $f(2)$ as the new x-value to find $f(f(2))$ again on the blue line.

Let me read the values from the graph to find these results.

Reading Values from the Graph:

• For (a):

  • $g(4) = -3$ (red line at $x = 4$)
  • $f(-3) = -3$ (blue line at $x = -3$)

  $(f \circ g)(4) = -3$

• For (b):

  • $f(3) = -2$ (blue line at $x = 3$)
  • $g(-2) = 2$ (red line at $x = -2$)

  $(g \circ f)(3) = 2$

• For (c):

  • $f(2) = -1$ (blue line at $x = 2$)
  • $f(-1) = -2$ (blue line at $x = -1$)

  $(f \circ f)(2) = -2$

Final Answers:

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

Would you like more details on any of these steps, or do you have any other questions?

Additional Related Questions:

1. How do you find the inverse of a function using its graph?
2. What does the composition of functions represent geometrically?
3. Can you find $(g \circ g)(x)$ for a different value of $x$ using the graph?
4. How can you determine if a function is linear or non-linear using its graph?
5. How do you calculate the slope of a linear function from its graph?

Tip:

When finding the value of a composition of functions, always start from the inner function and work your way out.