The domain of the definition of the function $f\left( x \right) = \frac{1}{{4 - {x^2}}} + \log \,\left( {{x^3} - x} \right)$ is

Show that the Modulus Function $f : R \rightarrow R$ given by $(x)=|x|$, is neither one - one nor onto, where $|x|$ is $x$, if $x$ is positive or $0$ and $| X |$ is $- x$, if $x$ is negative.

Let $f$ be a function defined on the set of all positive integers such that $f(x y)=f(x)+f(y)$ for all positive integers $x, y$. If $f(12)=24$ and $f(8)=15$. The value of $f(48)$ is

Let $f(x ) = x^3 - 2x + 2$. If real numbers $a$, $b$ and $c$ such that $\left| {f\left( a \right)} \right| + \left| {f\left( b \right)} \right| + \left| {f\left( c \right)} \right| = 0$ then the value of ${f^2}\left( {{a^2} + \frac{2}{a}} \right) + {f^2}\left( {{b^2} + \frac{2}{b}} \right) - {f^2}\left( {{c^2} + \frac{2}{c}} \right)$ equal to

Let $R$ be the set of real numbers and $f: R \rightarrow R$ be defined by $f(x)=\frac{\{x\}}{1+[x]^2}$, where $[x]$ is the greatest integer less than or equal to $x$, and $\left\{x{\}}=x-[x]\right.$. Which of the following statements are true?

$I.$ The range of $f$ is a closed interval.

$II.$ $f$ is continuous on $R$.

$III.$ $f$ is one-one on $R$

If $f:R \to R$ and $g:R \to R$ are given by $f(x) = \;|x|$ and $g(x) = \;|x|$ for each $x \in R$, then $\{ x \in R\;:g(f(x)) \le f(g(x))\} = $