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 how to determine the asymptote of an exponential function and the domain of a quadratic function through detailed examples of f(x) = -x^2 - 2x + 8 and g(x) = 2^{-x} + 2. Learn about their behavior as x approaches infinity and negative infinity. Suitable for high school students.

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