To determine the domain and range of the given exponential function, let’s analyze the properties of exponential functions and the values provided in the table:
1. Exponential functions are typically continuous and defined for all real numbers. Therefore, the domain of an exponential function should logically include all real numbers rather than just integers.
2. Analyzing the [tex]\( y \)[/tex]-values given in the table:
- 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]
These values suggest that [tex]\( y \)[/tex] is an exponential function of [tex]\( x \)[/tex].
3. For exponential functions of the form [tex]\( y = ab^x \)[/tex], the [tex]\( y \)[/tex]-values grow exponentially as [tex]\( x \)[/tex] increases. Since [tex]\( y = 4 \)[/tex] when [tex]\( x = 0 \)[/tex], we find the initial value of [tex]\( y \)[/tex] is at least 4. As [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] increases as well.
4. Since [tex]\( y = 4 \)[/tex] is a result when [tex]\( x = 0 \)[/tex], the minimum value [tex]\( y \)[/tex] can take based on this function appears to be 4. However, because exponential functions grow rapidly and become very large, the range should include all values greater than 4.
Given these points:
- The domain of the function should be the set of all real numbers.
- The range of the function starts at 4 and includes all numbers greater than 4, given that the initial value is 4 and the function increases thereafter.
Thus, the correct answer is:
The domain is the set of real numbers, and the range is [tex]\( y > 4 \)[/tex].
