To determine the domain and range of the given continuous exponential function, we need to analyze the provided data points: (0, 4), (1, 5), (2, 6.25), and (3, 7.8125).
Step-by-Step Analysis:
1. Domain Determination:
- In mathematics, the domain of a function is the set of all possible input values (x-values) that the function can accept.
- For a continuous exponential function, the domain is typically all real numbers because you can input any real number into an exponential function and receive a corresponding output.
- Thus, based on the nature of an exponential function, we conclude that the domain is the set of real numbers.
2. Range Determination:
- The range of a function is the set of all possible output values (y-values) that the function can produce.
- To understand the range, we observe the given data points and recognize the pattern:
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 5 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 6.25 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = 7.8125 \)[/tex]
- We see that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] also increases, and [tex]\( y \)[/tex] values are always greater than 4.
- Therefore, we can infer from the given data points that the possible [tex]\( y \)[/tex] values must be greater than 4.
- Consequently, the range is [tex]\( y > 4 \)[/tex].
Putting these together, the correct answers are:
- Domain: The set of real numbers.
- Range: [tex]\( y > 4 \)[/tex].
Thus, the accurate statement from the given choices is:
- The domain is the set of real numbers, and the range is [tex]\( y > 4 \)[/tex].
