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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.

Composition of Functions Using Tables of Values, evaluating expressions like f(g(6)), g(f(3)), f(f(0)), and g(g(1)) based on given function tables.

Learn how to evaluate the composition of functions f(g(x)), g(f(x)), f(f(x)), and g(g(x)) using tables of values. This problem walks through evaluating functions step-by-step, using tables to find the values of f(x) and g(x). Suitable for high school students in Grades 9-12.

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