To determine which function has a domain that contains the domain of the other as a subset, we first need to establish the domains of the given functions.
### Step 1: Domain of [tex]\( f(x) \)[/tex]
The function [tex]\( f(x) = -\log(x + 3) - 2 \)[/tex] involves a logarithmic expression [tex]\( \log(x + 3) \)[/tex]. For the logarithm to be defined, the argument inside the logarithm must be positive:
[tex]\[ x + 3 > 0 \][/tex]
Solving this inequality:
[tex]\[ x > -3 \][/tex]
Thus, the domain of [tex]\( f(x) \)[/tex] can be expressed in interval notation as:
[tex]\[ (-3, \infty) \][/tex]
### Step 2: Domain of [tex]\( g(x) \)[/tex]
Since we are not provided with an explicit function [tex]\( g(x) \)[/tex], we can assume a common generic function for comparison. A general and simple function to consider is [tex]\( g(x) = x \)[/tex], which is defined for all real numbers. Therefore, the domain of [tex]\( g(x) \)[/tex] is:
[tex]\[ (-\infty, \infty) \][/tex]
### Step 3: Comparing Domains
Now that we have the domains of both functions, we can compare them:
- The domain of [tex]\( f(x) \)[/tex] is [tex]\( (-3, \infty) \)[/tex].
- The domain of [tex]\( g(x) \)[/tex] is [tex]\( (-\infty, \infty) \)[/tex].
The domain of [tex]\( f(x) \)[/tex], which is [tex]\( (-3, \infty) \)[/tex], is entirely contained within the domain of [tex]\( g(x) \)[/tex], which is [tex]\( (-\infty, \infty) \)[/tex].
### Conclusion
The function [tex]\( g(x) = x \)[/tex] has a domain that contains the domain of [tex]\( f(x) = -\log(x + 3) - 2 \)[/tex] as a subset.
Therefore, the function, [tex]\( \boxed{g(x)} \)[/tex], has a domain that contains the domain of the other function as a subset.
