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We are given the function \( f(x) = x^2 + 4 \), and we need to find a function \( g(x) \) such that \( (f \circ g)(x) = x + 6 \). 

### Step-by-step approach:

1. **Recall that \( (f \circ g)(x) = f(g(x)) \)**. This means we need to substitute \( g(x) \) into the function \( f(x) \).
   
   Since \( f(x) = x^2 + 4 \), we can write:
   \[
   f(g(x)) = (g(x))^2 + 4
   \]
   
2. **Set this equal to the given condition**:
   \[
   (g(x))^2 + 4 = x + 6
   \]

3. **Solve for \( g(x) \)**:
   \[
   (g(x))^2 + 4 = x + 6
   \]
   Subtract 4 from both sides:
   \[
   (g(x))^2 = x + 2
   \]
   Now take the square root of both sides:
   \[
   g(x) = \pm \sqrt{x + 2}
   \]
   
Thus, the function \( g(x) \) can be \( g(x) = \sqrt{x + 2} \) or \( g(x) = -\sqrt{x + 2} \).

Would you like any further clarification or details on this solution?

### Related Questions:
1. What is the difference between \( f(g(x)) \) and \( g(f(x)) \)?
2. Can we find \( g(x) \) for different forms of \( f(x) \)?
3. What would happen if the equation for \( f(x) \) was more complex, like \( f(x) = x^3 + 4 \)?
4. What would be the inverse function of \( f(x) \) if it were invertible?
5. How would you solve \( f(g(x)) = x + c \) for any constant \( c \)?

### Tip:
When solving function composition problems, always start by recalling the definition \( (f \circ g)(x) = f(g(x)) \) to simplify the problem step by step.

The function f is given by f(x) = x^2 + 4. Find a function g such that (f ∘ g)(x) = x + 6.

Explore how to find the function g where f(g(x)) = x + 6 for the function f(x) = x^2 + 4. This step-by-step algebraic solution covers function composition and square roots, suitable for students in Grades 10-12.

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