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To find the function \( g(x) \), we need to analyze the given functions and equations:

Given:
\[ f(x) = \sqrt{2x + 6} \]
\[ (f \circ g)(x) = f(g(x)) = 4x^2 \]

To find \( g(x) \), we'll use the composition of functions.

1. Start by expressing \( f(g(x)) = 4x^2 \):
   \[ f(g(x)) = \sqrt{2g(x) + 6} \]

2. Set this equal to \( 4x^2 \):
   \[ \sqrt{2g(x) + 6} = 4x^2 \]

3. Square both sides to eliminate the square root:
   \[ 2g(x) + 6 = (4x^2)^2 \]

4. Simplify the right side:
   \[ 2g(x) + 6 = 16x^4 \]

5. Solve for \( g(x) \):
   \[ 2g(x) = 16x^4 - 6 \]
   \[ g(x) = 8x^4 - 3 \]

Thus, the function \( g(x) \) is:

\[
g(x) = 8x^4 - 3
\]

Would you like more details or have any questions about this solution? 

Here are some related questions to consider:

1. How would you verify that \( g(x) = 8x^4 - 3 \) is correct by substituting it back into the composition \( f(g(x)) \)?
2. What is the domain of the function \( g(x) = 8x^4 - 3 \)?
3. Can you find \( g(x) \) if \( f(x) = \sqrt{3x + 7} \) and \( (f \circ g)(x) = 9x^2 \)?
4. How would the solution change if the equation \( (f \circ g)(x) = f(g(x)) = 4x^2 + 1 \) was given instead?
5. Can you determine \( g(x) \) if \( f(x) = x^2 \) and \( (f \circ g)(x) = 16x^8 + 9 \)?

**Tip:** When dealing with composition functions, it's often helpful to work backwards from the inside out to determine the unknown function.

Learn how to find the function g(x) given f(x) = √(2x + 6) and (f ∘ g)(x) = 4x^2. Step-by-step solution using function composition and square root function. Suitable for advanced high school students and beyond.

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