To determine the domain and range of the function [tex]\( f(x) = 3^x + 5 \)[/tex], we'll analyze the properties of the function step-by-step:
### Domain
1. Definition of the Exponential Function:
The function [tex]\( 3^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means you can plug any real number into [tex]\( 3^x \)[/tex] and get a valid output.
2. Adding a Constant:
Adding the constant 5 to [tex]\( 3^x \)[/tex] doesn't change the fact that [tex]\( 3^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Given these points, the domain of [tex]\( f(x) = 3^x + 5 \)[/tex] is all real numbers:
[tex]\[ \text{Domain}: (-\infty, \infty) \][/tex]
### Range
1. Behavior of the Exponential Function:
The function [tex]\( 3^x \)[/tex] is an exponential function, and its range is [tex]\( (0, \infty) \)[/tex] because [tex]\( 3^x \)[/tex] is always positive for any real [tex]\( x \)[/tex] and can get arbitrarily large as [tex]\( x \)[/tex] increases and arbitrarily close to 0 as [tex]\( x \)[/tex] decreases.
2. Adding a Constant:
Adding 5 to [tex]\( 3^x \)[/tex] shifts the entire graph of [tex]\( 3^x \)[/tex] upwards by 5 units. This transformation affects the range as follows:
- The minimum value of [tex]\( 3^x \)[/tex] is just above 0 (when [tex]\( x \)[/tex] goes to [tex]\(-\infty\)[/tex]), so [tex]\( 3^x + 5 \)[/tex] will be just above 5.
- There is no upper bound to the values of [tex]\( 3^x + 5 \)[/tex] since [tex]\( 3^x \)[/tex] can become arbitrarily large.
Therefore, the range of [tex]\( f(x) = 3^x + 5 \)[/tex] is:
[tex]\[ \text{Range}: (5, \infty) \][/tex]
### Conclusion
Combining the domain and range, the correct option is:
[tex]\[ \text{Domain}: (-\infty, \infty) \][/tex]
[tex]\[ \text{Range}: (5, \infty) \][/tex]
So, the answer to the question is:
- [tex]\(\text{domain: } (-\infty, \infty)\)[/tex]
- [tex]\(\text{range: } (5, \infty)\)[/tex]
