Let's analyze the given function step-by-step to determine its domain and range.
Given:
[tex]\[ y = 4^{x-5} + 3 \][/tex]
### Domain:
The domain of a function consists of all possible input values (x-values) that the function can accept. In this case, [tex]\(4^{x-5}\)[/tex] is an exponential function, and exponential functions are defined for all real numbers. There are no restrictions on the value of [tex]\(x\)[/tex] that would make the expression undefined. Therefore, the domain of the function is all real numbers.
So, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
### Range:
To find the range, we analyze the output values (y-values) of the function.
1. Start with the exponential part, [tex]\(4^{x-5}\)[/tex]:
- Exponential functions with a positive base greater than 1 are always positive, so [tex]\(4^{x-5} > 0\)[/tex].
2. Adding 3 to the exponential term:
- Since [tex]\(4^{x-5}\)[/tex] is always positive, the smallest value [tex]\(4^{x-5}\)[/tex] can approach is 0 (as [tex]\(x \to -\infty\)[/tex]), but it never actually reaches 0. When we add 3 to it, the minimum value of [tex]\(y\)[/tex] is just slightly greater than 3.
- As [tex]\(x\)[/tex] increases, [tex]\(4^{x-5}\)[/tex] grows exponentially without bound, causing [tex]\(y\)[/tex] to increase without bound as well.
Combining these observations:
- The smallest value [tex]\(y\)[/tex] can approach is 3 (but [tex]\(y\)[/tex] will always be greater than 3).
- There is no upper limit to the value of [tex]\(y\)[/tex].
Hence, the range of [tex]\(y\)[/tex] is all real numbers greater than 3.
So, the range is:
[tex]\[ (3, \infty) \][/tex]
### Conclusion:
- The domain of this function is [tex]\(\boxed{(-\infty, \infty)}\)[/tex].
- The range of this function is [tex]\(\boxed{(3, \infty)}\)[/tex].
