To determine the domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex], let's carefully analyze the properties of the function.
### Analyzing the Function
1. Type of Function:
- The function [tex]\( f(x) = 5^x - 7 \)[/tex] is an exponential function where the base is 5 and the exponent is [tex]\( x \)[/tex].
- Since the base of the exponential function is a positive number greater than 1 (5 in this case), the exponential part [tex]\( 5^x \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
2. Transformations:
- The term [tex]\( -7 \)[/tex] is a vertical shift downward by 7 units of the basic exponential function [tex]\( 5^x \)[/tex].
- Vertical shifts do not affect the domain of the function; they merely move the graph of the function up or down along the y-axis.
### Determining the Domain
- Exponential functions of the form [tex]\( 5^x \)[/tex] are defined for every real number [tex]\( x \)[/tex] because you can raise 5 to any real number exponent.
- Subtracting 7 from [tex]\( 5^x \)[/tex] does not impose any additional restrictions on the values [tex]\( x \)[/tex] can take.
Thus, the function [tex]\( f(x) = 5^x - 7 \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
### Conclusion
The domain of the function [tex]\( f(x) = 5^x - 7 \)[/tex] is:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
So, the correct choice is:
[tex]\[ \boxed{\{ x \mid x \text{ is a real number} \}} \][/tex]
