Evaluate each expression using the graphs of y equals f left parenthesis x right parenthesis and y equals g left parenthesis x right parenthesis shown below.
a. f left parenthesis g left parenthesis 4 right parenthesis right parenthesis
b. g left parenthesis f left parenthesis 9 right parenthesis right parenthesis
c. f left parenthesis g left parenthesis 6 right parenthesis right parenthesis
d. g left parenthesis f left parenthesis 2 right parenthesis right parenthesis
e. f left parenthesis f left parenthesis 6 right parenthesis right parenthesis
f.  g left parenthesis f left parenthesis g left parenthesis 2 right parenthesis right parenthesis right parenthesis

To evaluate each expression using the graphs of \( y = f(x) \) and \( y = g(x) \), follow these steps:

1. **Find the inner function value**: For the expression \( f(g(4)) \), first find \( g(4) \) using the graph of \( y = g(x) \).
2. **Substitute the inner function value into the outer function**: Once you have \( g(4) \), substitute it into the function \( f(x) \) and find \( f(g(4)) \) using the graph of \( y = f(x) \).

Let’s go through each part:

### a. \( f(g(4)) \)
- First, find \( g(4) \) from the graph of \( y = g(x) \).
- Then, use the value of \( g(4) \) to find \( f(g(4)) \) from the graph of \( y = f(x) \).

### b. \( g(f(9)) \)
- First, find \( f(9) \) from the graph of \( y = f(x) \).
- Then, use the value of \( f(9) \) to find \( g(f(9)) \) from the graph of \( y = g(x) \).

### c. \( f(g(6)) \)
- First, find \( g(6) \) from the graph of \( y = g(x) \).
- Then, use the value of \( g(6) \) to find \( f(g(6)) \) from the graph of \( y = f(x) \).

### d. \( g(f(2)) \)
- First, find \( f(2) \) from the graph of \( y = f(x) \).
- Then, use the value of \( f(2) \) to find \( g(f(2)) \) from the graph of \( y = g(x) \).

### e. \( f(f(6)) \)
- First, find \( f(6) \) from the graph of \( y = f(x) \).
- Then, use the value of \( f(6) \) to find \( f(f(6)) \) from the graph of \( y = f(x) \).

### f. \( g(f(g(2))) \)
- First, find \( g(2) \) from the graph of \( y = g(x) \).
- Then, use the value of \( g(2) \) to find \( f(g(2)) \) from the graph of \( y = f(x) \).
- Finally, use the value of \( f(g(2)) \) to find \( g(f(g(2))) \) from the graph of \( y = g(x) \).

If you can provide the graphs or the corresponding values, I can help you find the exact values for each expression.

Would you like to share the graph, or do you have any questions so far?

### Related Questions
1. What is the value of \( f(4) \) from the graph of \( y = f(x) \)?
2. How do you interpret the composition of two functions graphically?
3. What does \( g(f(x)) \) represent in terms of function operations?
4. How do you find the inverse of a function using its graph?
5. What are the key differences between the graphs of \( f(x) \) and \( g(x) \)?

### Tip
When dealing with function compositions, always start with the innermost function and work your way outward to ensure accuracy.

Learn how to evaluate expressions using the graphs of functions f(x) and g(x). Includes step-by-step solutions for expressions like f(g(4)), g(f(9)), and more. Understand function composition and graphical analysis of functions. Suitable for advanced high school students and anyone interested in function operations.

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