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The image contains the following mathematical expression and instructions:

1. You are given:  
   \((g \circ f)(x) = 3x^2 + 4\)

2. It is also stated:  
   \(f(x) = x + 1\), and the task is to determine \(g(x)\).

We are tasked with finding the function \(g(x)\), given the composition \((g \circ f)(x)\) and \(f(x)\).

### Step-by-Step Solution:

1. The composition of functions \((g \circ f)(x)\) means \(g(f(x))\). This implies that:
   \[
   g(f(x)) = 3x^2 + 4
   \]
2. From the given, we know that \(f(x) = x + 1\). Substituting this into the expression for \(g(f(x))\):
   \[
   g(x + 1) = 3x^2 + 4
   \]

3. Now, to find \(g(x)\), let’s substitute \(y = x + 1\), which gives \(x = y - 1\). This transforms the equation into:
   \[
   g(y) = 3(y - 1)^2 + 4
   \]

4. Expand the expression on the right-hand side:
   \[
   g(y) = 3(y^2 - 2y + 1) + 4
   \]
   Simplify it:
   \[
   g(y) = 3y^2 - 6y + 3 + 4
   \]
   \[
   g(y) = 3y^2 - 6y + 7
   \]

Therefore, the function \(g(x)\) is:
\[
g(x) = 3x^2 - 6x + 7
\]

Let me know if you would like further details or clarifications!

Here are 5 related questions that expand on this topic:
1. How would the solution change if \(f(x)\) were a different function, say \(f(x) = 2x + 3\)?
2. What is the general procedure for finding the composition of two functions?
3. Can you solve for \(f(x)\) if \(g(x)\) and \((g \circ f)(x)\) are known?
4. How does the composition of functions relate to the chain rule in calculus?
5. What is the geometric interpretation of composing two functions?

**Tip:** When solving compositions, focus on expressing one function entirely in terms of the other to simplify the process.

Determine g(x) given (g ∘ f)(x) = 3x² + 4 and f(x) = x + 1.

Learn how to solve for g(x) in a composition of functions where (g ∘ f)(x) = 3x² + 4 and f(x) = x + 1. This problem covers composition of functions and quadratic functions, with a detailed step-by-step explanation. Suitable for high school students (Grades 10-12).

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