Answered

The graph of [tex]f(x)[/tex] is shown below.

If [tex]g(x)=0.5^{(x+1)}+9[/tex], which statement is true?

A. The horizontal asymptote of [tex]f(x)[/tex] is [tex]y=1[/tex], and the horizontal asymptote of [tex]g(x)[/tex] is [tex]y=9[/tex].
B. The horizontal asymptote of [tex]f(x)[/tex] is [tex]y=1[/tex], and the horizontal asymptote of [tex]g(x)[/tex] is [tex]y=1[/tex].
C. The horizontal asymptote of [tex]f(x)[/tex] is [tex]y=1[/tex], and the horizontal asymptote of [tex]g(x)[/tex] is [tex]y=-9[/tex].
D. The horizontal asymptote of [tex]f(x)[/tex] is [tex]y=1[/tex], and the horizontal asymptote of [tex]g(x)[/tex] is [tex]y=-1[/tex].



Answer :

GinnyAnswer
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]