Let's determine the domain and range of the function [tex]\( f(x) = \log(x + 6) - 4 \)[/tex].
### 1. Finding the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined.
For the function [tex]\( f(x) = \log(x + 6) - 4 \)[/tex]:
- The term inside the logarithm [tex]\( \log(x + 6) \)[/tex] must be greater than 0 because the logarithm is undefined for zero and negative numbers.
- Therefore, we need [tex]\( x + 6 > 0 \)[/tex].
Solving the inequality:
[tex]\[ x + 6 > 0 \][/tex]
[tex]\[ x > -6 \][/tex]
Thus, the domain of the function is [tex]\( x > -6 \)[/tex].
### 2. Finding the Range
The range of a function is the set of all possible output values (y-values).
For the function [tex]\( f(x) = \log(x + 6) - 4 \)[/tex]:
- The logarithm function [tex]\( \log(x + 6) \)[/tex] can output any real number because the logarithm of a positive number covers all real numbers.
- After taking the logarithm, subtracting 4 does not restrict the range but instead shifts the entire range of the logarithm function downward by 4 units.
- Therefore, the transformation [tex]\( \log(x + 6) - 4 \)[/tex] still covers all real numbers.
Thus, the range of the function is all real numbers.
### Conclusion
Hence, the correct domain and range of the function [tex]\( f(x) = \log(x + 6) - 4 \)[/tex] are:
- Domain: [tex]\( x > -6 \)[/tex]
- Range: All real numbers
So, the correct answer is:
Domain: [tex]\(x > -6 \)[/tex]; Range: all real numbers
