The problem in the image involves the transformation of the function $f(x) = 8^x$ through a series of steps. Let's break down each transformation.

Step 1: Vertical Stretch

The function is first multiplied by 4, which represents a vertical stretch by a factor of 4: $h(x) = 4 \cdot f(x) = 4 \cdot 8^x$

Step 2: Horizontal Translation

Next, the function $h(x)$ is translated 3 units to the left. This means that $x$ is replaced by $x + 3$: $g(x) = h(x + 3) = 4 \cdot 8^{x + 3}$

Step 3: Vertical Translation

Finally, the function is translated 1 unit up, which adds 1 to the result: $g(x) = 4 \cdot 8^{x + 3} + 1$

Thus, the final expression for $g(x)$ is: $g(x) = 4 \cdot 8^{x + 3} + 1$

This concludes the transformations: vertical stretch, horizontal translation, and vertical translation.

Would you like more details on any step, or do you have any other questions?

Here are some related questions to deepen your understanding:

1. What effect does a horizontal translation have on the graph of a function?
2. How does a vertical stretch affect the graph compared to a horizontal stretch?
3. What happens if you translate a function by a negative value horizontally or vertically?
4. How would the transformations change if the original function was $f(x) = 2^x$ instead of $8^x$?
5. Can you describe how combining multiple transformations (like stretch, translation, and reflection) changes the overall graph of a function?

Tip: Always apply horizontal transformations (translations or stretches) directly to $x$ by modifying the input variable, and vertical transformations (translations or stretches) to the entire function output.