To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], we need to understand where the function is defined.
1. Exponential Term [tex]\( 5^x \)[/tex]:
- The term [tex]\( 5^x \)[/tex] is an exponential expression with base 5.
- Exponential functions are defined for all real numbers. There are no restrictions on the value of [tex]\( x \)[/tex] for the expression [tex]\( 5^x \)[/tex].
2. Subtraction Term [tex]\(-7\)[/tex]:
- Subtracting 7 from [tex]\( 5^x \)[/tex] does not impose any additional restrictions on the domain. It simply shifts the output of the function vertically downward by 7 units.
Since there are no restrictions on the value of [tex]\( x \)[/tex] for the expression [tex]\( 5^x \)[/tex] and subtracting 7 also does not impose any restrictions, we conclude that the function [tex]\( f(x) = 5^x - 7 \)[/tex] is defined for all real numbers.
Conclusion:
The domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex] is all real numbers.
In set notation, the domain can be written as:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
Thus, the correct answer is:
\[ \{ x \mid x \text{ is a real number} \} \
