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The given function is:

\[
g(x) = 9^x + 3^{x+1} + 2 \cdot 3^x + 6
\]

We are asked to determine if the function has a minimum value, and if so, what the minimum value is.

### Step-by-step approach:

1. **Simplify the function**:
   \[
   g(x) = 9^x + 3^{x+1} + 2 \cdot 3^x + 6
   \]
   Recall that \(9^x = (3^2)^x = (3^x)^2\), and simplify the term \(3^{x+1} = 3^x \cdot 3\):
   
   \[
   g(x) = (3^x)^2 + 3 \cdot 3^x + 2 \cdot 3^x + 6
   \]
   
   Combine like terms:
   
   \[
   g(x) = (3^x)^2 + 5 \cdot 3^x + 6
   \]

2. **Substitute \(y = 3^x\)**, making the equation a quadratic form:
   \[
   g(x) = y^2 + 5y + 6
   \]
   where \(y = 3^x\) (and since \(3^x > 0\), \(y > 0\)).

3. **Find the critical points of the quadratic function**:
   The quadratic function \(g(y) = y^2 + 5y + 6\) is a parabola, and since the coefficient of \(y^2\) is positive, the parabola opens upwards. Thus, it has a minimum value at its vertex.

   The vertex of a quadratic function \(ay^2 + by + c\) occurs at \(y = -\frac{b}{2a}\). Here, \(a = 1\) and \(b = 5\):
   \[
   y = -\frac{5}{2 \cdot 1} = -\frac{5}{2}
   \]
   
   However, since \(y = 3^x > 0\), we are only interested in positive values of \(y\). The minimum value for \(y = 3^x\) occurs when \(y = 1\) (this corresponds to \(x = 0\)).

4. **Substitute \(y = 1\) back into the quadratic**:
   \[
   g(1) = 1^2 + 5 \cdot 1 + 6 = 1 + 5 + 6 = 12
   \]

Thus, the minimum value of the function \(g(x)\) occurs at \(x = 0\), and the minimum value is \(12\).

### Conclusion:
The function \(g(x)\) has a minimum value of 12, and this occurs when \(x = 0\).

Do you want further details or have any questions?

Here are some related questions to extend the topic:
1. What is the derivative of the function \(g(x)\) before substitution?
2. How can we determine the domain and range of \(g(x)\)?
3. What happens to the function \(g(x)\) as \(x \to \infty\)?
4. Can we graph the function \(g(x)\) to visually confirm the minimum?
5. How does the function change if we modify the constant term \(6\) in \(g(x)\)?

**Tip**: Recognizing patterns in exponential functions and their transformations can simplify complex algebraic manipulations.

Diberikan g(x) = 9^x + 3^{x+1} + 2 · 3^x + 6. Apakah fungsi g mempunyai nilai minimum? Jika ada, berapa nilai minimumnya?

This problem explores the exponential function g(x) = 9^x + 3^{x+1} + 2 · 3^x + 6. Learn how to simplify the function into a quadratic form and calculate its minimum value using the vertex formula for parabolas. Suitable for high school students studying advanced algebra and calculus.

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