Let's analyze the problem and find the details step-by-step.

Problem Summary:

1. We start with the function $f(x) = 8^x$.
2. We need to transform this function to $g(x) = 8^{-x}$.
3. The question also asks to determine the equation of the asymptote for $g(x)$ and identify the domain and range.

Analysis:

1. Transformation Identification:

   • The function $g(x) = 8^{-x}$ can be thought of as a reflection of $f(x) = 8^x$ across the y-axis. This is because replacing $x$ with $-x$ reflects the graph horizontally.
   • From the options given, the correct choice is C: The graph of $f(x) = 8^x$ should be reflected about the y-axis.

2. Asymptote:

   • The function $g(x) = 8^{-x}$ is an exponential decay function. As $x \to \infty$, $8^{-x} \to 0$.
   • Therefore, the horizontal asymptote for $g(x)$ is $y = 0$.

3. Domain and Range:

   • Domain: For exponential functions like $g(x) = 8^{-x}$, $x$ can take any real value. Thus, the domain is $(-\infty, \infty)$.
   • Range: Since $g(x) = 8^{-x}$ is always positive (never zero), the range is $(0, \infty)$.

Summary of Answers:

• Transformation: Reflection about the y-axis.
• Asymptote: $y = 0$.
• Domain: $(-\infty, \infty)$.
• Range: $(0, \infty)$.

Do you want more detailed explanations or have any questions?

Here are some related questions to explore further:

1. What happens to the range if we add a constant to $g(x) = 8^{-x}$?
2. How would the transformation change if we used $g(x) = -8^x$ instead?
3. What would the asymptote be if we considered $g(x) = 8^{x+1}$?
4. How does the base of an exponential function affect the rate of growth or decay?
5. Can we determine the y-intercept of $g(x) = 8^{-x}$?

Tip: For exponential functions, replacing $x$ with $-x$ always results in a reflection about the y-axis.