To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], we need to investigate for which values of [tex]\( x \)[/tex] the function is defined.
1. Exponential Function Analysis:
- The base of the exponential function here is 5, which is a positive real number.
- The exponential function [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This means there are no restrictions on the value of [tex]\( x \)[/tex]; you can input any real number, and [tex]\( 5^x \)[/tex] will result in a defined output.
2. Subtraction:
- The function is [tex]\( 5^x - 7 \)[/tex]. Since [tex]\( 5^x \)[/tex] is defined for all real numbers, subtracting 7 doesn't change this fact.
- The subtraction of a constant (in this case, 7) from an already defined value of [tex]\( 5^x \)[/tex] does not impose any new restrictions on [tex]\( x \)[/tex].
Hence, [tex]\( f(x) = 5^x - 7 \)[/tex] is defined for all real numbers. There are no values of [tex]\( x \)[/tex] that would make [tex]\( f(x) \)[/tex] undefined.
Therefore, the domain of [tex]\( f(x) = 5^x - 7 \)[/tex] is all real numbers.
Thus, the correct choice is:
[tex]\[
\{x \mid x \text{ is a real number} \}
\][/tex]
