What are the domain and range of the function [tex][tex]$f(x) = 2^{x+1}$[/tex][/tex]?

A. Domain: [tex][tex]$(0, \infty)$[/tex][/tex]; Range: [tex][tex]$(-\infty, \infty)$[/tex][/tex]
B. Domain: [tex][tex]$(-\infty, \infty)$[/tex][/tex]; Range: [tex][tex]$(0, \infty)$[/tex][/tex]
C. Domain: [tex][tex]$(-\infty, \infty)$[/tex][/tex]; Range: [tex][tex]$(2, \infty)$[/tex][/tex]
D. Domain: [tex][tex]$(0, \infty)$[/tex][/tex]; Range: [tex][tex]$(2, \infty)$[/tex][/tex]



Answer :

To determine the domain and range of the function [tex]\( f(x) = 2^{x+1} \)[/tex], we need to analyze the behavior and properties of exponential functions.

### Domain

For the given function [tex]\( f(x) = 2^{x+1} \)[/tex]:

1. Exponential Functions Definition:
- Exponential functions are defined for all real numbers. This means that no matter what value of [tex]\( x \)[/tex] we choose, [tex]\( 2^{x+1} \)[/tex] is always defined and results in a real number.

2. Conclusion for Domain:
- Therefore, the domain of [tex]\( f(x) = 2^{x+1} \)[/tex] includes all real numbers.

[tex]\[ \text{Domain} = (-\infty, \infty) \][/tex]

### Range

For the given function [tex]\( f(x) = 2^{x+1} \)[/tex]:

1. Behavior of Exponential Functions:
- The base of the exponential function here is 2, which is greater than 1. Exponential growth functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a > 1 \)[/tex]) always yield positive results for any real number [tex]\( x \)[/tex].

2. Positive Real Output:
- The function [tex]\( 2^{x+1} \)[/tex] will never yield zero or negative values. As [tex]\( x \)[/tex] gets very large (positive), [tex]\( 2^{x+1} \)[/tex] approaches infinity. As [tex]\( x \)[/tex] gets very small (negative), [tex]\( 2^{x+1} \)[/tex] approaches zero but never actually reaches zero.

3. Conclusion for Range:
- Thus, the range of [tex]\( f(x) = 2^{x+1} \)[/tex] includes all positive real numbers.

[tex]\[ \text{Range} = (0, \infty) \][/tex]

### Final Answer

[tex]\[ \text{Domain:} \quad (-\infty, \infty) \][/tex]
[tex]\[ \text{Range:} \quad (0, \infty) \][/tex]

So, the correct domain and range for the function [tex]\( f(x) = 2^{x+1} \)[/tex] are:

[tex]\[ \boxed{\text{Domain: } (-\infty, \infty),\; \text{Range: } (0, \infty)} \][/tex]