Let's analyze the function [tex]\( y = 4.5 \left(\frac{1}{2}\right)^x + 8 \)[/tex] step-by-step.
### Domain:
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the given function [tex]\( y = 4.5 \left(\frac{1}{2}\right)^x + 8 \)[/tex]:
- The base function [tex]\(\left(\frac{1}{2}\right)^x\)[/tex] is defined for all real numbers [tex]\(x\)[/tex].
- Multiplying by 4.5 and adding 8 does not impose any additional restrictions.
Therefore, the domain of the function is all real numbers.
### Range:
The range of a function is the set of all possible output values (y-values) that the function can produce.
For the given function [tex]\( y = 4.5 \left(\frac{1}{2}\right)^x + 8 \)[/tex]:
- First, consider the exponential part [tex]\( 4.5 \left(\frac{1}{2}\right)^x \)[/tex]. As [tex]\(x \to \infty\)[/tex], [tex]\(\left(\frac{1}{2}\right)^x \to 0\)[/tex], and consequently [tex]\(4.5 \left(\frac{1}{2}\right)^x \to 0\)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\(\left(\frac{1}{2}\right)^x \to \infty\)[/tex], but since [tex]\(0 < \frac{1}{2} < 1\)[/tex], it means that [tex]\(4.5 \left(\frac{1}{2}\right)^x \)[/tex] also becomes very small, approaching 0 but never becoming negative.
Thus, [tex]\(4.5 \left(\frac{1}{2}\right)^x\)[/tex] is always positive and approaches 0 as [tex]\(x\)[/tex] increases.
When adding 8 to the minimum value of [tex]\( 4.5 \left(\frac{1}{2}\right)^x \)[/tex], which is approaching 0 from the positive side:
- The smallest value of [tex]\( y \)[/tex] approaches but is always greater than [tex]\( 8 \)[/tex].
Thus, the range is all real numbers [tex]\( y \)[/tex] such that [tex]\( y > 8 \)[/tex].
### Conclusion:
Domain: All Real Numbers
Range: [tex]\( y > 8 \)[/tex]
Based on this detailed analysis, the correct answer from the provided options is:
Domain: All Real Numbers
Range: [tex]\( y > 8 \)[/tex]
