To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], we need to identify all the possible values that [tex]\( x \)[/tex] can take.
1. The function [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. This is because the exponential function [tex]\( a^x \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]) is defined for all real values of [tex]\( x \)[/tex].
2. Subtracting 7 from [tex]\( 5^x \)[/tex] does not change the fact that [tex]\( 5^x \)[/tex] can be computed for any real number [tex]\( x \)[/tex]. The subtraction merely shifts the graph of the exponential function down by 7 units.
Since there are no restrictions on [tex]\( x \)[/tex] in the definition of [tex]\( 5^x \)[/tex] or in the operation of subtracting 7, [tex]\( x \)[/tex] can be any real number. Hence, the domain of the function is all real numbers.
Therefore, the domain of [tex]\( f(x) = 5^x - 7 \)[/tex] is:
[tex]\[ \{x \mid x \text{ is a real number} \} \][/tex]
