To determine the domain, range, and asymptote of the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex], let’s break it down step-by-step.
1. Domain:
The domain of a function is all the possible values that [tex]\( x \)[/tex] can take. In the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex], there are no restrictions on [tex]\( x \)[/tex] because [tex]\( (0.5)^x \)[/tex] is defined for all real numbers. Thus, the domain includes all real numbers.
[tex]\[
\text{Domain: } \{ x \mid x \text{ is a real number} \}
\][/tex]
2. Range:
The range of a function is all the possible values that [tex]\( h(x) \)[/tex] can take. The term [tex]\( (0.5)^x \)[/tex] represents an exponential decay function, which yields positive values for all real [tex]\( x \)[/tex]. As [tex]\( x \to -\infty \)[/tex], [tex]\( (0.5)^x \)[/tex] approaches [tex]\( +\infty \)[/tex]; and as [tex]\( x \to \infty \)[/tex], [tex]\( (0.5)^x \)[/tex] approaches [tex]\( 0 \)[/tex]. Subtracting 9 from [tex]\( (0.5)^x \)[/tex] shifts the entire graph down by 9 units.
This means that the lowest value [tex]\( h(x) \)[/tex] can approach is just slightly below -9 (but never reaching -9). Therefore, [tex]\( h(x) \)[/tex] will always be greater than -9.
[tex]\[
\text{Range: } \{ y \mid y > -9 \}
\][/tex]
3. Asymptote:
An asymptote of a function is a line that the graph of the function approaches but never touches. For the function [tex]\( h(x) = (0.5)^x - 9 \)[/tex], as [tex]\( x \to \infty \)[/tex], [tex]\( (0.5)^x \)[/tex] approaches 0. Therefore, the function [tex]\( h(x) \)[/tex] approaches [tex]\( -9 \)[/tex]. This indicates a horizontal asymptote at [tex]\( y = -9 \)[/tex].
[tex]\[
\text{Asymptote: } y = -9
\][/tex]
Now, let's match our results with the given choices:
- Domain: [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex]
- Range: [tex]\( \{ y \mid y > -9 \} \)[/tex]
- Asymptote: [tex]\( y = -9 \)[/tex]
The correct answer matches our findings:
[tex]\[
\boxed{\text{domain: } \{ x \mid x \text{ is a real number} \}; \text{ range: } \{ y \mid y > -9 \}; \text{ asymptote: } y = -9}
\][/tex]
