To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], we need to identify all possible values that [tex]\( x \)[/tex] can take for the function to be defined.
1. Understanding the Exponential Function:
- The given function is [tex]\( f(x) = 5^x - 7 \)[/tex].
- The primary component is the exponential expression [tex]\( 5^x \)[/tex].
2. Domain of [tex]\( 5^x \)[/tex]:
- The exponential function [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex]. There are no restrictions on the values that [tex]\( x \)[/tex] can take because an exponential function does not have any undefined points or discontinuities. Therefore, [tex]\( 5^x \)[/tex] is defined for all real numbers.
3. Effect of the Subtraction:
- The function [tex]\( f(x) = 5^x - 7 \)[/tex] involves subtracting 7 from [tex]\( 5^x \)[/tex].
- Subtracting a constant (which is 7 in this case) from [tex]\( 5^x \)[/tex] does not affect the domain of the exponential function.
- Thus, the subtraction only shifts the graph of [tex]\( 5^x \)[/tex] vertically but does not introduce any new restrictions on the values of [tex]\( x \)[/tex].
4. Conclusion:
- Therefore, the domain of [tex]\( f(x) = 5^x - 7 \)[/tex] is the same as the domain of [tex]\( 5^x \)[/tex].
- Since [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex], [tex]\( f(x) = 5^x - 7 \)[/tex] is also defined for all real numbers [tex]\( x \)[/tex].
Thus, the correct answer is:
[tex]\[ \{x \mid x \text{ is a real number} \} \][/tex]
Therefore, the domain of [tex]\( f(x) = 5^x - 7 \)[/tex] is all real numbers.
