To identify the domain of combined functions involving two functions \( f(x) \) and \( g(x) \), we need to consider the individual domains of \( f \) and \( g \). The domain of a combined function is generally determined by the most restrictive domain between \( f \) and \( g \) as well as any restrictions due to the operations being performed.
Let's denote the domains of \( f(x) \) and \( g(x) \) as \( D_f \) and \( D_g \), respectively.
1. Domain of \( (f+g)(x) \)
The function \( (f+g)(x) \) is defined wherever both \( f(x) \) and \( g(x) \) are defined. Thus, the domain of \( (f+g)(x) \) can be expressed as:
\[ D_{(f+g)} = D_f \cap D_g \]
2. Domain of \( (f-g)(x) \)
Similar to addition, the subtraction operation requires both functions to be defined. Therefore, the domain of \( (f-g)(x) \) is the same as for \( (f+g)(x) \):
\[ D_{(f-g)} = D_f \cap D_g \]
3. Domain of \( (f \cdot g)(x) \)
For multiplication, both functions must be defined. Hence, the domain of \( (f \cdot g)(x) \) is also:
\[ D_{(f \cdot g)} = D_f \cap D_g \]
4. Domain of \( \frac{f}{g}(x) \)
When considering division, we must ensure that \( g(x) \) is not equal to zero alongside both \( f(x) \) and \( g(x) \) being defined. Therefore, the domain of \( \frac{f}{g}(x) \) is given by:
\[ D_{\left(\frac{f}{g}\right)} = D_f \cap D_g \cap { x \mid g(x) \neq 0 } \]
Summary:
- \( D_{(f+g)} = D_f \cap D_g \)
- \( D_{(f-g)} = D_f \cap D_g \)
- \( D_{(f \cdot g)} = D_f \cap D_g \)
- \( D_{\left(\frac{f}{g}\right)} = D_f \cap D_g \cap { x \mid g(x) \neq 0 } \)
Note: In practice, the specific domains of \( f(x) \) and \( g(x) \) will need to be provided to identify the final numerical domain for these combined functions. Remember to always check for specific values or intervals that could further restrict your combined function domains.