To find the common domain of the functions \( f+g \), \( f-g \), \( \frac{f}{g} \), and \( \frac{g}{f} \) where \( f(x) = \sqrt{x} \) and \( g(x) = \sqrt{1-x} \), we will analyze the domain of each function step by step.
### Step 1: Determine the domain of \( f(x) = \sqrt{x} \)
The function \( f(x) = \sqrt{x} \) is defined when the expression inside the square root is non-negative:
\[
x \geq 0
\]
Thus, the domain of \( f \) is:
\[
D_f = [0, \infty)
\]
### Step 2: Determine the domain of \( g(x) = \sqrt{1-x} \)
The function \( g(x) = \sqrt{1-x} \) is defined when the expression inside the square root is non-negative:
\[
1 - x \geq 0 \implies x \leq 1
\]
Thus, the domain of \( g \) is:
\[
D_g = (-\infty, 1]
\]
### Step 3: Find the common domain for \( f+g \)
The domain of \( f+g \) is the intersection of the domains of \( f \) and \( g \):
\[
D_{f+g} = D_f \cap D_g = [0, \infty) \cap (-\infty, 1] = [0, 1]
\]
### Step 4: Find the common domain for \( f-g \)
Similarly, the domain of \( f-g \) is also the intersection of the domains of \( f \) and \( g \):
\[
D_{f-g} = D_f \cap D_g = [0, \infty) \cap (-\infty, 1] = [0, 1]
\]
### Step 5: Find the common domain for \( \frac{f}{g} \)
For \( \frac{f}{g} \) to be defined, \( g \) must not be zero:
\[
g(x) \neq 0 \implies \sqrt{1-x} \neq 0 \implies 1 - x \neq 0 \implies x \neq 1
\]
Thus, the domain of \( \frac{f}{g} \) is:
\[
D_{\frac{f}{g}} = D_f \cap D_g \cap \{ x \neq 1 \} = [0, \infty) \cap (-\infty, 1] \cap \{ x \neq 1 \} = [0, 1)
\]
### Step 6: Find the common domain for \( \frac{g}{f} \)
For \( \frac{g}{f} \) to be defined, \( f \) must not be zero:
\[
f(x) \neq 0 \implies \sqrt{x} \neq 0 \implies x \neq 0
\]
Thus, the domain of \( \frac{g}{f} \) is:
\[
D_{\frac{g}{f}} = D_f \cap D_g \cap \{ x \neq 0 \} = [0, \infty) \cap (-\infty, 1] \cap \{ x \neq 0 \} = (0, 1]
\]
### Step 7: Find the common domain of all functions
Now we find the intersection of all the domains:
\[
D = D_{f+g} \cap D_{f-g} \cap D_{\frac{f}{g}} \cap D_{\frac{g}{f}} = [0, 1] \cap [0, 1] \cap [0, 1) \cap (0, 1]
\]
This results in:
\[
D = (0, 1)
\]
### Final Answer
The common domain of \( f+g \), \( f-g \), \( \frac{f}{g} \), and \( \frac{g}{f} \) is:
\[
(0, 1)
\]
