https://file.aiquickdraw.com/mathbot/file/172738086810513e6lexx.jpg

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.

Which of the following expressions has a value of 2: (f ∘ g)(-3), (g ∘ f)(-3), (f ∘ g)(0), or (g ∘ f)(0)?

Learn how to evaluate function compositions with a graph and quadratic functions. We solve expressions such as (f ∘ g)(-3) and (g ∘ f)(0), using graph interpretation for f(x) and the quadratic function g(x) = (x + 1)^2. Suitable for Grades 9-11.

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