Answer :
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]
