Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.
Sagot :
To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], we need to analyze where this function is defined.
1. Exponential Function Analysis:
- The function [tex]\( f(x) \)[/tex] includes an exponential term [tex]\( 5^x \)[/tex].
- Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex].
2. Translation and Domain:
- The function [tex]\( f(x) = 5^x - 7 \)[/tex] is obtained by subtracting 7 from the exponential function [tex]\( 5^x \)[/tex].
- Translating a function vertically by subtracting or adding a constant does not affect the domain. It only changes the range.
3. Conclusion on Domain:
- Since the base exponential function [tex]\( 5^x \)[/tex] is defined for all real numbers, subtracting 7 still allows the function to be defined for all real numbers.
- Therefore, [tex]\( f(x) = 5^x - 7 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are real numbers.
Thus, the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex] is:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
The correct choice is:
[tex]\[ \boxed{4} \][/tex]
1. Exponential Function Analysis:
- The function [tex]\( f(x) \)[/tex] includes an exponential term [tex]\( 5^x \)[/tex].
- Exponential functions of the form [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) are defined for all real numbers [tex]\( x \)[/tex].
2. Translation and Domain:
- The function [tex]\( f(x) = 5^x - 7 \)[/tex] is obtained by subtracting 7 from the exponential function [tex]\( 5^x \)[/tex].
- Translating a function vertically by subtracting or adding a constant does not affect the domain. It only changes the range.
3. Conclusion on Domain:
- Since the base exponential function [tex]\( 5^x \)[/tex] is defined for all real numbers, subtracting 7 still allows the function to be defined for all real numbers.
- Therefore, [tex]\( f(x) = 5^x - 7 \)[/tex] is defined for all values of [tex]\( x \)[/tex] that are real numbers.
Thus, the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex] is:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
The correct choice is:
[tex]\[ \boxed{4} \][/tex]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.