Functions

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 – 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:

Let $$f:\mathbb{R}\to\mathbb{R}$$ be a function defined by $$f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\}$$, for some $$m$$, such that the range of $$f$$ is [0, 2]. Then the value of $$m$$ 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

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?

Let x denote the total number of one-one functions from a set A with 3 elements to a set B with 5 elements and y denote the total number of one-one functions form the set A to the set A $$\times$$ B. Then :

Let $$A = \{ 1,2,3,....,10\}$$ and $$f:A \to A$$ be defined as f(k) = \left\{ {\matrix{ {k + 1} & {if\,k\,is\,odd} \cr k & {if\,k\,is\,even} \cr } } \right. Then the number of possible functions $$g:A \to A$$ such that $$gof = f$$ is :

Let g : N $$\to$$ N be defined as g(3n + 1) = 3n + 2, g(3n + 2) = 3n + 3, g(3n + 3) = 3n + 1, for all n $$\ge$$ 0. Then which of the following statements is true?

Let a function f : N $$\to$$ N be defined by f(n) = \left[ {\matrix{ {2n,} & {n = 2,4,6,8,......} \cr {n - 1,} & {n = 3,7,11,15,......} \cr {{{n + 1} \over 2},} & {n = 1,5,9,13,......} \cr } } \right. then, f is