To determine the domain and range of the exponential function [tex]\( F(x) = 3^x + 2 \)[/tex], let's analyze the function step-by-step.
### Step 1: Determine the Domain
The domain of a function consists of all the possible input values (x-values) for which the function is defined. An exponential function of the form [tex]\( 3^x \)[/tex] is defined for all real numbers because for any real number [tex]\( x \)[/tex], [tex]\( 3^x \)[/tex] will produce a valid output.
Adding 2 to [tex]\( 3^x \)[/tex] does not change the domain, so the domain of the function [tex]\( F(x) = 3^x + 2 \)[/tex] is:
[tex]\[ \text{All real numbers} \][/tex]
### Step 2: Determine the Range
The range of a function consists of all possible output values (y-values) the function can take.
1. Start by examining the simpler function [tex]\( G(x) = 3^x \)[/tex]. The exponential function [tex]\( 3^x \)[/tex] produces outputs which are always positive. As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 3^x \)[/tex] approaches 0 from the positive side. As [tex]\( x \)[/tex] increases, [tex]\( 3^x \)[/tex] increases rapidly without bound. Hence, the range of [tex]\( 3^x \)[/tex] is:
[tex]\[ (0, \infty) \][/tex]
2. Now consider the function [tex]\( F(x) = 3^x + 2 \)[/tex]. Adding 2 to [tex]\( 3^x \)[/tex] shifts the entire graph of [tex]\( 3^x \)[/tex] upwards by 2 units. Therefore, the smallest value [tex]\( 3^x \)[/tex] can get close to is 0, making the smallest value [tex]\( 3^x + 2 \)[/tex] can get close to is 2.
Since [tex]\( 3^x + 2 \)[/tex] never actually reaches 2 but it can get arbitrarily close to 2, and it can also increase without bound as [tex]\( x \)[/tex] increases, the range of the function [tex]\( F(x) = 3^x + 2 \)[/tex] is:
[tex]\[ (2, \infty) \][/tex]
In other words:
[tex]\[ \text{All real numbers greater than 2} \][/tex]
### Conclusion
Given our analysis:
- The Domain of [tex]\( F(x) = 3^x + 2 \)[/tex] is all real numbers.
- The Range of [tex]\( F(x) = 3^x + 2 \)[/tex] is all real numbers greater than 2.
Thus, the correct answer is:
D. Domain: All real numbers Range: All real numbers greater than 2
