Given functions f(x)= -x²-2x+8 and g(x)=$$ 2^{-x}+2 $$ Write down the asymptote of g(x) and determine the domain of f(x)

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.

Explore the asymptote of g(x) = 2^{-x} + 2 and the domain of f(x) = -x^2 - 2x + 8 with detailed step-by-step solutions. This problem covers key concepts in quadratic and exponential functions, including asymptotes and function domains. Suitable for high school students.

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