Let's analyze the given information and determine which statement is true regarding the horizontal asymptotes of the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
### Horizontal Asymptote of [tex]\( f(x) \)[/tex]
The horizontal asymptote of [tex]\( f(x) \)[/tex] is given explicitly in the problem statement. We already know that:
[tex]\[ \text{The horizontal asymptote of } f(x) \text{ is } y = 1. \][/tex]
### Horizontal Asymptote of [tex]\( g(x) \)[/tex]
Now, let's find the horizontal asymptote of [tex]\( g(x) = 0.5^{(x+1)} + 9 \)[/tex].
1. Consider the function [tex]\( g(x) = 0.5^{(x+1)} + 9 \)[/tex].
2. As [tex]\( x \)[/tex] approaches infinity ([tex]\( x \to \infty \)[/tex]):
- The term [tex]\( 0.5^{(x+1)} \)[/tex] becomes very small because any number raised to an increasingly large power where the base is less than 1 approaches 0.
- Therefore, [tex]\( 0.5^{(x+1)} \)[/tex] will approach 0.
This means that as [tex]\( x \to \infty \)[/tex]:
[tex]\[ g(x) \approx 0 + 9 = 9. \][/tex]
Thus, the horizontal asymptote of [tex]\( g(x) \)[/tex] is:
[tex]\[ y = 9. \][/tex]
### Conclusion
Given both findings:
- The horizontal asymptote of [tex]\( f(x) \)[/tex] is [tex]\( y = 1 \)[/tex].
- The horizontal asymptote of [tex]\( g(x) \)[/tex] is [tex]\( y = 9 \)[/tex].
Therefore, the correct statement is:
[tex]\[ \boxed{A. \text{The horizontal asymptote of } f(x) \text{ is } y = 1, \text{ and the horizontal asymptote of } g(x) \text{ is } y = 9.} \][/tex]
