https://file.aiquickdraw.com/mathbot/file/1726809734327a1xe08lz.jpg

From the provided image, we have two graphs representing functions \( f(x) \) and \( g(x) \). We need to compute the following:

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

Let's break these down step by step using the graphs:

### (a) \( g(f(1)) \):
1. First, find \( f(1) \) from the graph of \( f(x) \). 
   - Looking at the graph, when \( x = 1 \), \( f(1) = 3 \).
2. Now, substitute \( f(1) = 3 \) into \( g(x) \), i.e., find \( g(3) \).
   - From the graph of \( g(x) \), when \( x = 3 \), \( g(3) = -2 \).

Thus, \( g(f(1)) = g(3) = -2 \).

### (b) \( f(g(2)) \):
1. First, find \( g(2) \) from the graph of \( g(x) \).
   - From the graph, when \( x = 2 \), \( g(2) = 0 \).
2. Now, substitute \( g(2) = 0 \) into \( f(x) \), i.e., find \( f(0) \).
   - From the graph of \( f(x) \), when \( x = 0 \), \( f(0) = 5 \).

Thus, \( f(g(2)) = f(0) = 5 \).

### Final Results:
- \( g(f(1)) = -2 \)
- \( f(g(2)) = 5 \)

Would you like further details or explanations? Here are five related questions to deepen understanding:

1. How do we interpret function composition when given two graphs?
2. What is the domain and range of the given functions \( f \) and \( g \)?
3. How would \( f(g(x)) \) change if we changed the graph's scale?
4. What are some applications of function composition in real-world problems?
5. Could you solve \( g(f(x)) \) for other values of \( x \)?

**Tip**: Always check the axes' scales carefully when interpreting graphs to avoid mistakes in reading values.

Use the graphs of f and g shown below to find the following: (a) g(f(1)) ≈ ? (b) f(g(2)) ≈ ?

Learn how to compute function compositions like g(f(1)) and f(g(2)) using graphical data. This problem involves understanding function composition, reading graphs, and interpreting points on function curves. Suitable for students in Grades 9-12.

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