Let's go through each part of the problem by interpreting the composition of functions from the table.

Given table values:

• $x$ values: $-3, -2, -1, 0, 1, 2, 3$
• $f(x)$: $10, 8, 6, 4, 2, 0, -2$
• $g(x)$: $-9, -3, 0, 3, 0, -3, -9$

Solutions:

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

1. Find $g(1)$: From the table, $g(1) = 0$.
2. Then, find $f(g(1)) = f(0)$: From the table, $f(0) = 4$.
3. So, $(f \circ g)(1) = 4$.

b. $(f \circ g)(2)$

1. Find $g(2)$: From the table, $g(2) = -3$.
2. Then, find $f(g(2)) = f(-3)$: From the table, $f(-3) = 10$.
3. So, $(f \circ g)(2) = 10$.

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

1. Find $f(2)$: From the table, $f(2) = 0$.
2. Then, find $g(f(2)) = g(0)$: From the table, $g(0) = 3$.
3. So, $(g \circ f)(2) = 3$.

d. $(g \circ f)(3)$

1. Find $f(3)$: From the table, $f(3) = -2$.
2. Then, find $g(f(3)) = g(-2)$: From the table, $g(-2) = -3$.
3. So, $(g \circ f)(3) = -3$.

e. $(g \circ g)(1)$

1. Find $g(1)$: From the table, $g(1) = 0$.
2. Then, find $g(g(1)) = g(0)$: From the table, $g(0) = 3$.
3. So, $(g \circ g)(1) = 3$.

f. $(f \circ f)(3)$

1. Find $f(3)$: From the table, $f(3) = -2$.
2. Then, find $f(f(3)) = f(-2)$: From the table, $f(-2) = 8$.
3. So, $(f \circ f)(3) = 8$.

Summary of Answers:

• a. $(f \circ g)(1) = 4$
• b. $(f \circ g)(2) = 10$
• c. $(g \circ f)(2) = 3$
• d. $(g \circ f)(3) = -3$
• e. $(g \circ g)(1) = 3$
• f. $(f \circ f)(3) = 8$

Would you like further details on any of these solutions?

Here are 5 related questions to expand your understanding:

1. How do you interpret function composition in general terms?
2. Can you explain the difference between $(f \circ g)(x)$ and $(g \circ f)(x)$?
3. What are some real-world applications of function composition?
4. How does function composition affect the domain and range of functions?
5. Can you find any other compositions of functions using this table?

Tip: When evaluating compositions, always start from the innermost function and work outward.