Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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]
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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Discover more at Westonci.ca. Return for the latest expert answers and updates on various topics.