Functions

The number of bijective functions $f:\{1,3,5,7, \ldots, 99\} \rightarrow\{2,4,6,8, \ldots .100\}$, such that $f(3) \geq f(9) \geq f(15) \geq f(21) \geq \ldots . . f(99)$, is ____________.

The domain of ${\sin ^{ - 1}}\left[ {{{\log }_3}\left( {{x \over 3}} \right)} \right]$ is

Let a – 2b + c = 1. If f(x)=\left| {\matrix{ {x + a} & {x + 2} & {x + 1} \cr {x + b} & {x + 3} & {x + 2} \cr {x + c} & {x + 4} & {x + 3} \cr } } \right|, then:

The range of $f(x)=4 \sin ^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)$ is

Let $f(x)=\frac{1}{7-\sin 5 x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to :

Let [ x ] denote the greatest integer $\le$ x, where x $\in$ R. If the domain of the real valued function $f(x) = \sqrt {{{\left| {[x]} \right| - 2} \over {\left| {[x]} \right| - 3}}}$ is ($-$ $\infty$, a) $]\cup$ [b, c) $\cup$ [4, $\infty$), a < b < c, then the value of a + b + c is :

A function $f$ from the set of natural numbers to integers defined by $f\left( n \right) = \left\{ {\matrix{ {{{n - 1} \over 2},\,when\,n\,is\,odd} \cr { - {n \over 2},\,when\,n\,is\,even} \cr } } \right.$ is

If $f:R \to S$, defined by $f\left( x \right) = \sin x - \sqrt 3 \cos x + 1$, is onto, then the interval of $S$ is

Let $f:( - 1,1) \to B$, be a function defined by $f\left( x \right) = {\tan ^{ - 1}}{{2x} \over {1 - {x^2}}}$, then $f$ is both one-one and onto when B is the interval

Let $f:N \to Y$ be a function defined as f(x) = 4x + 3 where Y = { y $\in$ N, y = 4x + 3 for some x $\in$ N }. Show that f is invertible and its inverse is

Let $f\left( x \right) = {\left( {x + 1} \right)^2} - 1,x \ge - 1$ Statement - 1 : The set $\left\{ {x:f\left( x \right) = {f^{ - 1}}\left( x \right)} \right\} = \left\{ {0, - 1} \right\}$. Statement - 2 : $f$ is a bijection.

For real x, let f(x) = x³ + 5x + 1, then

The function $f:R \to \left[ { - {1 \over 2},{1 \over 2}} \right]$ defined as $f\left( x \right) = {x \over {1 + {x^2}}}$, is

The function f : N $\to$ N defined by f (x) = x $-$ 5 $\left[ {{x \over 5}} \right],$ Where N is the set of natural numbers and [x] denotes the greatest integer less than or equal to x, is :

Let A = {x $\in$ R : x is not a positive integer}. Define a function $f$ : A $\to$  R   as  $f(x)$ = ${{2x} \over {x - 1}}$, then $f$ is :

The number of functions f from {1, 2, 3, ...., 20} onto {1, 2, 3, ...., 20} such that f(k) is a multiple of 3, whenever k is a multiple of 4, is :

Let a function f : (0, $\infty$) $\to$ (0, $\infty$) be defined by f(x) = $\left| {1 - {1 \over x}} \right|$. Then f is :

If the function ƒ : R – {1, –1} $\to$ A defined by ƒ(x) = ${{{x^2}} \over {1 - {x^2}}}$ , is surjective, then A is equal to

Let f, g : N $\to$ N such that f(n + 1) = f(n) + f(1) $\forall$ n$\in$N and g be any arbitrary function. Which of the following statements is NOT true?