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.