Which statement best describes the domain and range of [tex]f(x) = -(7)^x[/tex] and [tex]g(x) = 7^x[/tex]?

A. [tex]f(x)[/tex] and [tex]g(x)[/tex] have the same domain and the same range.
B. [tex]f(x)[/tex] and [tex]g(x)[/tex] have the same domain but different ranges.
C. [tex]f(x)[/tex] and [tex]g(x)[/tex] have different domains but the same range.
D. [tex]f(x)[/tex] and [tex]g(x)[/tex] have different domains and different ranges.



Answer :

Let's analyze the functions \( f(x) = -(7)^x \) and \( g(x) = 7^x \) in terms of their domains and ranges.

### Domain
The domain of a function is the set of all possible input values (x-values) that the function can accept.

1. For \( f(x) = -(7)^x \):
- The base 7 raised to any real number \( x \) is defined, so \( 7^x \) is defined for all real numbers \( x \).
- Multiplying by -1 does not affect the domain.
- Therefore, the domain of \( f(x) = -(7)^x \) is all real numbers.

2. For \( g(x) = 7^x \):
- Similarly, \( 7^x \) is defined for any real number \( x \).
- Therefore, the domain of \( g(x) = 7^x \) is also all real numbers.

Conclusion: Both \( f(x) \) and \( g(x) \) have the same domain, which is all real numbers.

### Range
The range of a function is the set of all possible output values (y-values) that the function can produce.

3. For \( f(x) = -(7)^x \):
- \( 7^x \) for any real number \( x \) is always positive (greater than 0).
- Multiplying by -1 flips the positive values to their negative counterparts.
- Therefore, the range of \( f(x) = -(7)^x \) is all negative real numbers.

4. For \( g(x) = 7^x \):
- \( 7^x \) for any real number \( x \) is always positive (greater than 0).
- Therefore, the range of \( g(x) = 7^x \) is all positive real numbers.

Conclusion: The ranges of \( f(x) \) and \( g(x) \) are different. \( f(x) = -(7)^x \) covers all negative real numbers, while \( g(x) = 7^x \) covers all positive real numbers.

### Final Conclusion
Based on the above analysis, the correct statement is:

[tex]\[ \text{\[tex]$f(x)\$[/tex] and \[tex]$g(x)\$[/tex] have the same domain but different ranges.} \][/tex]