https://file.aiquickdraw.com/mathbot/file/1728615786461ncsvaj8i.jpg

The question asks you to use the graphs of \( f(x) \) (in blue) and \( g(x) \) (in red) to find the following values:

1. \( f(g(3)) \)
2. \( g(f(1)) \)
3. \( g(g(10)) \)

Let's walk through the process to solve these.

### 1. \( f(g(3)) \)

- **Step 1:** Find \( g(3) \) from the graph of \( g(x) \) (the red graph).
  - Look at the red graph at \( x = 3 \). From the image, it seems that \( g(3) = 8 \).

- **Step 2:** Now find \( f(8) \) from the graph of \( f(x) \) (the blue graph).
  - Look at the blue graph at \( x = 8 \). The value of \( f(8) \) appears to be approximately \( 0 \).

So, \( f(g(3)) = f(8) = 0 \).

### 2. \( g(f(1)) \)

- **Step 1:** Find \( f(1) \) from the graph of \( f(x) \).
  - Look at the blue graph at \( x = 1 \). The value of \( f(1) = 3 \).

- **Step 2:** Now find \( g(3) \) from the red graph.
  - From earlier, we know that \( g(3) = 8 \).

So, \( g(f(1)) = g(3) = 8 \).

### 3. \( g(g(10)) \)

- **Step 1:** Find \( g(10) \) from the red graph.
  - Look at the red graph at \( x = 10 \). The value of \( g(10) = 12 \).

- **Step 2:** Now find \( g(12) \) from the red graph.
  - Look at the red graph at \( x = 12 \). The value of \( g(12) = 0 \).

So, \( g(g(10)) = g(12) = 0 \).

### Summary of results:
1. \( f(g(3)) = 0 \)
2. \( g(f(1)) = 8 \)
3. \( g(g(10)) = 0 \)

Let me know if you'd like more details or clarifications!

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Here are 5 questions to extend your understanding:
1. What would \( f(f(5)) \) be using the same graphs?
2. How does the behavior of \( f(x) \) differ from \( g(x) \) over the interval \( [0, 10] \)?
3. Can you identify any points where both \( f(x) \) and \( g(x) \) share the same value for different \( x \)-values?
4. How would the result change if \( f(x) \) or \( g(x) \) were shifted up or down?
5. What is the domain and range of both \( f(x) \) and \( g(x) \) based on the graphs?

**Tip:** Always ensure that you read graphs carefully, as slight changes in values can lead to significant differences in results.

Use the figures above, which show the functions f(x) (blue graph) and g(x) (red graph), to find the following values: 1. f(g(3)) 2. g(f(1)) 3. g(g(10)).

Learn how to solve composite functions by using the graphs of f(x) and g(x). This problem demonstrates how to find values of f(g(x)), g(f(x)), and g(g(x)) by interpreting graphs, ideal for high school students in Grades 9-12.

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