If the functions are defined as $f(x) = \sqrt{x}$ and $g(x) = \sqrt{1-x}$,then what is the common domain of the following functions: $f+g, f-g, f/g, g/f, g-f$ where $(f \pm g)(x) = f(x) \pm g(x)$ and $(f/g)(x) = \frac{f(x)}{g(x)}$?

The domain of the function $f(x) = \sqrt{\log_{0.5} x!}$ is

Let $R$ be the relation on $Z$ defined by $R = \{(a, b) : a, b \in Z, a - b \text{ is an integer}\}$. Find the domain and range of $R$.

The range of the function $f(x) = \frac{x}{1 + |x|}, x \in R,$ is

Let the domain of the function $f(x) = \log_3 \log_5 (7 - \log_2 (x^2 - 10 x + 85)) + \sin^{-1} ( | \frac{3 x - 7}{17 - x} | )$ be $(\alpha, \beta]$. Then $\alpha + \beta$ is equal to :

Let $f(x)=\sqrt{2-x-x^2}$ and $g(x)=\cos x$. Which of the following statements are true?
$I$. Domain of $f((g(x))^2) = \text{Domain of } f(g(x))$
$II$. Domain of $f(g(x)) + g(f(x)) = \text{Domain of } g(f(x))$
$III$. Domain of $f(g(x)) = \text{Domain of } g(f(x))$
$IV$. Domain of $g((f(x))^3) = \text{Domain of } f(g(x))$