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]
### 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]
