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]
