Let's break down the table-based function composition problem step by step.

Functions given:

• $f(x)$ values are provided in the left table.
• $g(x)$ values are provided in the right table.

We need to evaluate the following compositions:

1. $f(g(6))$
2. $g(f(3))$
3. $f(f(0))$
4. $g(g(1))$

1. $f(g(6))$

• First, find $g(6)$ from the table for $g(x)$.
  • From the table: $g(6) = -1$.
• Now, substitute $g(6) = -1$ into $f(x)$, i.e., find $f(-1)$.
  • From the table: $f(-1) = 8$.
• So, $f(g(6)) = f(-1) = 8$.

2. $g(f(3))$

• First, find $f(3)$ from the table for $f(x)$.
  • From the table: $f(3) = 5$.
• Now, substitute $f(3) = 5$ into $g(x)$, i.e., find $g(5)$.
  • From the table: $g(5) = 7$.
• So, $g(f(3)) = g(5) = 7$.

3. $f(f(0))$

• First, find $f(0)$ from the table for $f(x)$.
  • From the table: $f(0) = -1$.
• Now, substitute $f(0) = -1$ into $f(x)$, i.e., find $f(-1)$.
  • From the table: $f(-1) = 8$.
• So, $f(f(0)) = f(-1) = 8$.

4. $g(g(1))$

• First, find $g(1)$ from the table for $g(x)$.
  • From the table: $g(1) = 6$.
• Now, substitute $g(1) = 6$ into $g(x)$, i.e., find $g(6)$.
  • From the table: $g(6) = -1$.
• So, $g(g(1)) = g(6) = -1$.

Final Answers:

1. $f(g(6)) = 8$
2. $g(f(3)) = 7$
3. $f(f(0)) = 8$
4. $g(g(1)) = -1$

Would you like more details or have any questions?

5 Related Questions:

1. How do you evaluate compositions of more complex functions?
2. Can you compose functions using their algebraic expressions instead of tables?
3. What is the domain and range of a composed function like $f(g(x))$?
4. How does the order of composition $f(g(x))$ vs. $g(f(x))$ affect the result?
5. How can function composition be applied to real-world problems?

Tip:

When composing functions using tables, always find the inner function's value first and then apply the result to the outer function.