To determine the domain, range, and asymptotes of the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex], we need to analyze the behavior and properties of the function step by step.
1. Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex], there are no restrictions on the values that [tex]\( x \)[/tex] can take. Exponential functions are always defined for all real numbers. Hence, the domain is:
[tex]\[
\{x \mid x \text{ is a real number}\}
\][/tex]
2. Range:
The range of a function is the set of all possible output values (y-values).
Consider the function [tex]\( g(x) = (0.5)^x \)[/tex]. The base of the exponential function is [tex]\(0.5\)[/tex], which is a positive number less than 1. As [tex]\( x \)[/tex] becomes very large, [tex]\( (0.5)^x \)[/tex] approaches 0 but never actually reaches 0. As [tex]\( x \)[/tex] becomes very large negative, [tex]\( (0.5)^x \)[/tex] increases without bound. So, [tex]\( g(x) \)[/tex] takes positive values.
Now, subtracting 9 from [tex]\( g(x) \)[/tex]:
[tex]\[
h(x) = (0.5)^x - 9
\][/tex]
This function will shift [tex]\( g(x) \)[/tex] vertically downward by 9 units. Therefore, the minimum value of [tex]\( h(x) \)[/tex] approaches [tex]\(-9\)[/tex] when [tex]\( x \)[/tex] goes to infinity, but [tex]\( h(x) \)[/tex] will never actually reach [tex]\(-9\)[/tex]. Consequently, for very large negative [tex]\( x \)[/tex], [tex]\( h(x) \)[/tex] can take values as large as we want. Thus, the range of [tex]\( h(x) \)[/tex] is:
[tex]\[
\{y \mid y > -9\}
\][/tex]
3. Asymptote:
An asymptote is a line that the graph of the function approaches but never touches or crosses. Given that [tex]\( (0.5)^x \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex], [tex]\( h(x) \)[/tex] will approach [tex]\( -9 \)[/tex] but never actually reach it. Thus, the horizontal asymptote of [tex]\( h(x) \)[/tex] is:
[tex]\[
y = -9
\][/tex]
Based on this analysis, we conclude that the domain, range, and horizontal asymptote of the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex] are as follows:
- Domain: [tex]\(\{x \mid x \text{ is a real number}\}\)[/tex]
- Range: [tex]\(\{y \mid y > -9\}\)[/tex]
- Asymptote: [tex]\( y = -9 \)[/tex]
Therefore, the correct choice from the given options is:
[tex]\[
\{x \mid x \text{ is a real number}\}; \{y \mid y > -9\}; y = -9
\][/tex]
