To solve these problems, we need to evaluate the compositions of functions $f$, $g$, and $h$ based on the graphs provided. Let's go through each question step-by-step using the graphs to find the values.

Given Graphs:

1. $f(x)$: Top graph
2. $g(x)$: Middle graph
3. $h(x)$: Bottom graph

Let's evaluate each composition:

19. $(f \circ g)(2)$

1. Find $g(2)$ from the graph of $g(x)$:

   • $g(2) = -3$.

2. Now, find $f(-3)$ from the graph of $f(x)$:

   • $f(-3) = 5$.

   Answer: $(f \circ g)(2) = 5$.

20. $f(g(1))$

1. Find $g(1)$ from the graph of $g(x)$:

   • $g(1) = 0$.

2. Now, find $f(0)$ from the graph of $f(x)$:

   • $f(0) = -2$.

   Answer: $f(g(1)) = -2$.

21. $g(h(-2))$

1. Find $h(-2)$ from the graph of $h(x)$:

   • $h(-2) = 0$.

2. Now, find $g(0)$ from the graph of $g(x)$:

   • $g(0) = -1$.

   Answer: $g(h(-2)) = -1$.

22. $(h \circ h)(-3)$

1. Find $h(-3)$ from the graph of $h(x)$:

   • $h(-3) = 1$.

2. Now, find $h(1)$ from the graph of $h(x)$:

   • $h(1) = 3$.

   Answer: $(h \circ h)(-3) = 3$.

23. $(h \circ f)(-2)$

1. Find $f(-2)$ from the graph of $f(x)$:

   • $f(-2) = 1$.

2. Now, find $h(1)$ from the graph of $h(x)$:

   • $h(1) = 3$.

   Answer: $(h \circ f)(-2) = 3$.

24. $g(g(-3))$

1. Find $g(-3)$ from the graph of $g(x)$:

   • $g(-3) = 1$.

2. Now, find $g(1)$ from the graph of $g(x)$:

   • $g(1) = 0$.

   Answer: $g(g(-3)) = 0$.

25. $g(h(f(2)))$

1. Find $f(2)$ from the graph of $f(x)$:

   • $f(2) = 2$.

2. Find $h(2)$ from the graph of $h(x)$:

   • $h(2) = -2$.

3. Now, find $g(-2)$ from the graph of $g(x)$:

   • $g(-2) = 4$.

   Answer: $g(h(f(2))) = 4$.

26. $g(f(h(0)))$

1. Find $h(0)$ from the graph of $h(x)$:

   • $h(0) = -2$.

2. Find $f(-2)$ from the graph of $f(x)$:

   • $f(-2) = 1$.

3. Now, find $g(1)$ from the graph of $g(x)$:

   • $g(1) = 0$.

   Answer: $g(f(h(0))) = 0$.

Would you like a deeper explanation for any of these steps, or do you have any other questions?

Related Questions

1. How do you interpret function compositions when given graphs?
2. How would you approach solving compositions if the function values were not given graphically?
3. Can you explain the difference between $(f \circ g)(x)$ and $f(g(x))$ in function notation?
4. What are the key steps to evaluate nested functions systematically?
5. How does domain and range affect evaluating compositions of functions?

Tip:

When evaluating compositions, always start from the innermost function and work outward for accurate results.