Thank you for pointing that out. Let's go over the calculations again carefully:

1. $f(g(1))$

• First, find $g(1)$ using the graph of $g(x)$.
  • From the graph, $g(1) = 2$.
• Then, find $f(2)$ using the graph of $f(x)$.
  • From the graph, $f(2) = 3$.
• Thus, $f(g(1)) = 3$.

2. $g(f(4))$

• As previously calculated:
  • $f(4) = 1$
  • $g(1) = 2$
• Thus, $g(f(4)) = 2$.

3. $f(f(2))$

• First, find $f(2)$ using the graph of $f(x)$.
  • From the graph, $f(2) = 3$.
• Then, find $f(3)$ using the graph of $f(x)$.
  • From the graph, $f(3) = 0$.
• Thus, $f(f(2)) = 0$.

4. $g(g(5))$

• First, find $g(5)$ using the graph of $g(x)$.
  • From the graph, $g(5) = 4$.
• Then, find $g(4)$ using the graph of $g(x)$.
  • From the graph, $g(4) = 2$.
• Thus, $g(g(5)) = 2$.

It seems the first three calculations are correct according to the analysis. Could you please confirm if there was any specific error I missed?

Would you like to review any specific steps?

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Related Questions:

1. How do function compositions work in a mathematical context?
2. How can we graph the composition of two functions?
3. What are some real-world examples where function composition is applied?
4. How does the range of one function affect the domain of its composition with another function?
5. Can all functions be composed together? What are the conditions?

Tip: Always double-check the values you obtain from the graph, as misreading a point can lead to incorrect function evaluations.