The problem asks for the end behavior of the exponential function $f(x) = 3^{-x} + 2$ as $x$ approaches infinity.

Step-by-Step Solution:

1. Analyze the function: $f(x) = 3^{-x} + 2$ The function consists of two parts: $3^{-x}$ and the constant $2$.

2. Behavior of $3^{-x}$ as $x$ approaches infinity: $3^{-x} = \frac{1}{3^x}$ As $x$ approaches infinity, $3^x$ becomes very large, so $\frac{1}{3^x}$ approaches 0.

3. Determine the end behavior of $f(x)$:

   • As $x$ approaches infinity, $3^{-x}$ approaches 0.
   • Therefore, $f(x)$ approaches $0 + 2 = 2$.

Conclusion:

The correct end behavior of $f(x)$ as $x$ approaches infinity is $f(x) \rightarrow 2$.

You selected the correct answer: $f(x) \rightarrow 2$.

Would you like any further details or have any questions? Here are five related questions:

1. What is the end behavior of the function $f(x) = 5^{-x} + 1$?
2. How does the base of the exponent affect the end behavior of exponential functions?
3. What is the horizontal asymptote of $f(x) = 2^{-x} + 4$?
4. How would the end behavior change if the function was $f(x) = 3^{x} + 2$ instead?
5. Can you describe the transformation applied to the basic exponential function $f(x) = 3^{-x}$ to obtain $f(x) = 3^{-x} + 2$?

Tip: When analyzing exponential functions, the behavior at infinity is typically dominated by the constant term, which usually defines the horizontal asymptote.