From the image provided, let’s analyze and solve each question based on the graph given. Here's the step-by-step explanation:

1. Graph analysis:
   • $f(x)$: Black line (thicker).
   • $g(x)$: Red line (thinner).

We’ll evaluate each question step by step:

a. $f(g(5))$

1. Find $g(5)$:

   • Look at the red line ($g(x)$) when $x = 5$.
   • At $x = 5$, $g(5) = 4$.

2. Find $f(4)$:

   • Now, look at the black line ($f(x)$) when $x = 4$.
   • At $x = 4$, $f(4) = 5$.

Answer: $f(g(5)) = 5$.

b. $g(f(2))$

1. Find $f(2)$:

   • Look at the black line ($f(x)$) when $x = 2$.
   • At $x = 2$, $f(2) = 3$.

2. Find $g(3)$:

   • Now, look at the red line ($g(x)$) when $x = 3$.
   • At $x = 3$, $g(3) = 2$.

Answer: $g(f(2)) = 2$.

c. $f(f(1))$

1. Find $f(1)$:

   • Look at the black line ($f(x)$) when $x = 1$.
   • At $x = 1$, $f(1) = 4$.

2. Find $f(4)$:

   • Now, look at the black line ($f(x)$) when $x = 4$.
   • At $x = 4$, $f(4) = 5$.

Answer: $f(f(1)) = 5$.

d. $g(g(0))$

1. Find $g(0)$:

   • Look at the red line ($g(x)$) when $x = 0$.
   • At $x = 0$, $g(0) = 1$.

2. Find $g(1)$:

   • Now, look at the red line ($g(x)$) when $x = 1$.
   • At $x = 1$, $g(1) = 4$.

Answer: $g(g(0)) = 4$.

Final Answers:

a. $f(g(5)) = 5$
b. $g(f(2)) = 2$
c. $f(f(1)) = 5$
d. $g(g(0)) = 4$

Would you like a detailed breakdown of any specific step or concept?

Expansion Questions:

1. What are composite functions, and how are they evaluated?
2. How do you read and interpret graphs of functions like $f(x)$ and $g(x)$?
3. Can you explain the difference between $f(g(x))$ and $g(f(x))$?
4. What are the domain and range of the functions $f(x)$ and $g(x)$ in this graph?
5. How does the vertical line test help us verify whether a graph represents a function?

Tip:

Always verify the corresponding $y$-value on the graph carefully when evaluating functions at specific $x$-values to avoid mistakes!