Let's analyze the functions [tex]\( f(x) = -(7)^x \)[/tex] and [tex]\( g(x) = 7^x \)[/tex] to determine their domain and range.
1. Domain Analysis:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] are exponential functions involving the base 7.
- Exponential functions are defined for all real numbers since exponentiation can be performed on any real number [tex]\( x \)[/tex].
Therefore, the domain for both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is all real numbers.
2. Range Analysis:
- For [tex]\( g(x) = 7^x \)[/tex]:
- Exponential functions with a positive base always yield positive results.
- As [tex]\( x \)[/tex] varies from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex], [tex]\( g(x) \)[/tex] moves from 0 (approaching but never reaching) to [tex]\(\infty\)[/tex].
- Therefore, the range of [tex]\( g(x) = 7^x \)[/tex] is all positive real numbers.
- For [tex]\( f(x) = -(7)^x \)[/tex]:
- This function is the negative of an exponential function.
- As [tex]\( 7^x \)[/tex] is always positive for all real [tex]\( x \)[/tex], multiplying by -1 will give all negative values.
- Thus, as [tex]\( x \)[/tex] varies from [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex], [tex]\( f(x) \)[/tex] moves from 0 (approaching but never reaching) to [tex]\(-\infty\)[/tex].
- Therefore, the range of [tex]\( f(x) = -(7)^x \)[/tex] is all negative real numbers.
In conclusion:
- Both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same domain: all real numbers.
- [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have different ranges: [tex]\( f(x) \)[/tex] has all negative real numbers, while [tex]\( g(x) \)[/tex] has all positive real numbers.
Thus, the best statement to describe the domain and range of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] is:
[tex]\[ \boxed{f(x) \text{ and } g(x) \text{ have the same domain but different ranges.}} \][/tex]
