Let's solve the problem step by step.

1. Asymptote of $g(x) = 2^{-x} + 2$

To find the asymptote, observe how $g(x)$ behaves as $x$ approaches infinity or negative infinity:

• As $x$ approaches infinity: $2^{-x}$ approaches 0 since the exponent becomes increasingly negative, making the value of $2^{-x}$ smaller and smaller.

  Thus, $g(x) = 2^{-x} + 2$ approaches $2$.

• As $x$ approaches negative infinity: $2^{-x}$ increases significantly because the exponent becomes increasingly positive. However, since $g(x)$ contains a constant $+2$, the value of $g(x)$ will be dominated by $2^{-x}$ and keep increasing without bound.

Given this, the horizontal asymptote of $g(x)$ is: $\text{Asymptote: } y = 2$

2. Domain of $f(x) = -x^2 - 2x + 8$

The function $f(x) = -x^2 - 2x + 8$ is a quadratic function, which is defined for all real values of $x$.

Therefore, the domain of $f(x)$ is: $\text{Domain: } (-\infty, \infty)$

Summary:

• Asymptote of $g(x)$: $y = 2$
• Domain of $f(x)$: $(-\infty, \infty)$

Would you like more details or have any questions?

Here are five related questions:

1. What is the range of $f(x) = -x^2 - 2x + 8$?
2. Can you find the vertex of the parabola defined by $f(x)$?
3. What is the range of $g(x) = 2^{-x} + 2$?
4. How would the asymptote change if the function $g(x)$ were $3^{-x} + 1$?
5. What are the roots of the quadratic function $f(x)$?

Tip: Remember, horizontal asymptotes describe the behavior of a function as $x$ goes to infinity or negative infinity, while the domain tells us all possible input values for the function.