To determine the equation of the horizontal asymptote for the function [tex]\( f(x) = \left(\frac{1}{2}\right)^x + 3 \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches infinity and negative infinity.
1. Understanding the function:
The function [tex]\( f(x) \)[/tex] has two parts:
- [tex]\( \left(\frac{1}{2}\right)^x \)[/tex]
- The constant term [tex]\( +3 \)[/tex]
2. Behavior of [tex]\( \left(\frac{1}{2}\right)^x \)[/tex] as [tex]\( x \)[/tex] increases:
- As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]), [tex]\( \left(\frac{1}{2}\right)^x \)[/tex] gets closer and closer to 0 because the base [tex]\( \frac{1}{2} \)[/tex] is less than 1.
- Therefore, for large values of [tex]\( x \)[/tex], [tex]\( \left(\frac{1}{2}\right)^x \approx 0 \)[/tex].
3. Adding the constant term [tex]\( +3 \)[/tex]:
- For large [tex]\( x \)[/tex], [tex]\( f(x) = \left(\frac{1}{2}\right)^x + 3 \approx 0 + 3 = 3 \)[/tex].
4. Horizontal asymptote:
- A horizontal asymptote is a horizontal line that the graph of the function approaches as [tex]\( x \)[/tex] goes to infinity or negative infinity.
- From the analysis above, as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to 3 \)[/tex]. Thus, the function approaches the horizontal line [tex]\( y = 3 \)[/tex].
Given this detailed analysis, the correct answer is:
D. [tex]\( y = 3 \)[/tex]
