Let's analyze the problem and solve it step by step.

The graph represents the function $f(x)$ defined on the interval $[-2, 9]$, and function $g(x)$ is given as:

$g(x) = (x + 1)^2$

We need to determine which of the given compositions results in a value of 2. The choices involve compositions of $f$ and $g$, denoted as $(f \circ g)(x)$ or $(g \circ f)(x)$, meaning we evaluate one function inside the other.

Let's solve each option.

Option A: $(f \circ g)(-3)$

This means we first evaluate $g(-3)$ and then $f$ at that result.

$g(-3) = (-3 + 1)^2 = (-2)^2 = 4$ Now, we need to find $f(4)$. From the graph, $f(4) = 2$.

So, $(f \circ g)(-3) = 2$.

Thus, Option A is correct.

Let's verify the other options:

Option B: $(g \circ f)(-3)$

This means we first evaluate $f(-3)$, but from the graph, $f(x)$ is only defined from $-2$ to $9$. So, this is not possible.

Option C: $(f \circ g)(0)$

First, evaluate $g(0)$: $g(0) = (0 + 1)^2 = 1$ Now, find $f(1)$ from the graph. $f(1) = 4$, so $(f \circ g)(0) = 4$, not 2.

Option D: $(g \circ f)(0)$

First, evaluate $f(0)$. From the graph, $f(0) = 2$.

Now, find $g(2)$: $g(2) = (2 + 1)^2 = 3^2 = 9$ So, $(g \circ f)(0) = 9$, not 2.

Conclusion:

The correct answer is Option A: $(f \circ g)(-3)$.

Would you like more details on any part of this solution? Here are 5 related questions to explore:

1. How is function composition defined mathematically?
2. What are the domain restrictions for composite functions like $(f \circ g)(x)$?
3. How do you interpret function graphs to find values of $f(x)$?
4. Can $g(x) = (x+1)^2$ have negative values for real $x$?
5. What are some common properties of quadratic functions like $g(x)$?

Tip: When working with composite functions, always check the range of the inner function and ensure it's within the domain of the outer function.