If domain of function $f(x) = \sqrt {\ln \left( {m\sin x + 4} \right)} $ is $R$ , then number of possible integral values of $m$ is
$3$
$4$
$6$
$7$
If $a, b$ be two fixed positive integers such that $f(a + x) = b + {[{b^3} + 1 - 3{b^2}f(x) + 3b{\{ f(x)\} ^2} - {\{ f(x)\} ^3}]^{\frac{1}{3}}}$ for all real $x$, then $f(x)$ is a periodic function with period
If $f(x) = \frac{x}{{x - 1}}$, then $\frac{{f(a)}}{{f(a + 1)}} = $
The domain of the function $f(x){ = ^{16 - x}}{\kern 1pt} {C_{2x - 1}}{ + ^{20 - 3x}}{\kern 1pt} {P_{4x - 5}}$, where the symbols have their usual meanings, is the set
Consider the identity function $I _{ N }: N \rightarrow N$ defined as $I _{ N }$ $(x)=x$ $\forall $ $x \in N$ Show that although $I _{ N }$ is onto but $I _{ N }+ I _{ N }:$ $ N \rightarrow N$ defined as $\left(I_{N}+I_{N}\right)(x)=$ $I_{N}(x)+I_{N}(x)$ $=x+x=2 x$ is not onto.
Set $A$ has $3$ elements and set $B$ has $4$ elements. The number of injection that can be defined from $A$ to $B$ is