The domain of the function $\log _{10}(x^2-5x+6)$ 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 $R$.

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 $g((f(x))^3) = \text{Domain of } f(g(x))$

If the domain of the function $f(x) = \log_{(10x^{2}-17x+7)}(18x^{2}-11x+1)$ is $(-\infty, a) \cup (b, c) \cup (d, \infty) - \{e\}$, then $90(a+b+c+d+e)$ equals:

The domain of the function $f(x) = \log_{3+x}(x^2 - 1)$ is

The domain of the real-valued function $f(x) = \frac{\sqrt{|x|-x}}{\sqrt{x-[x]}}$ is