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

For $$x \in \mathbb{R}$$, two real valued functions $$f(x)$$ and $$g(x)$$ are such that, $$g(x)=\sqrt{x}+1$$ and $$f \circ g(x)=x+3-\sqrt{x}$$. Then $...

Let $$\mathrm{D}$$ be the domain of the function $$f(x)=\sin ^{-1}\left(\log _{3 x}\left(\frac{6+2 \log _{3} x}{-5 x}\right)\right)$$. If the range of...

The domain of the function $$f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}$$ is : ( where $$[\mathrm{x}]$$ denotes the greatest integer less than or equal to ...

If $$f(x) = {{(\tan 1^\circ )x + {{\log }_e}(123)} \over {x{{\log }_e}(1234) - (\tan 1^\circ )}},x > 0$$, then the least value of $$f(f(x)) + f\left( ...

Let the sets A and B denote the domain and range respectively of the function $$f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}$$, where $$\lceil x\rceil$$ den...

Let $$f:\mathbb{R}-{0,1}\to \mathbb{R}$$ be a function such that $$f(x)+f\left(\frac{1}{1-x}\right)=1+x$$. Then $$f(2)$$ is equal to

Let $$f(x) = \left| {\matrix{ {1 + {{\sin }^2}x} & {{{\cos }^2}x} & {\sin 2x} \cr {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\sin 2x} \cr {{{...

Let $f: \mathbb{R}-\{2,6\} \rightarrow \mathbb{R}$ be real valued function defined as $f(x)=\frac{x^2+2 x+1}{x^2-8 x+12}$. Then range of $f$ is...

The absolute minimum value, of the function $f(x)=\left|x^{2}-x+1\right|+\left[x^{2}-x+1\right]$, where $[t]$ denotes the greatest integer function, ...

If the domain of the function $$f(x)=\frac{[x]}{1+x^{2}}$$, where $$[x]$$ is greatest integer $$\leq x$$, is $$[2,6)$$, then its range is

The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is :

Consider a function $$f:\mathbb{N}\to\mathbb{R}$$, satisfying $$f(1)+2f(2)+3f(3)+....+xf(x)=x(x+1)f(x);x\ge2$$ with $$f(1)=1$$. Then $$\frac{1}{f(2022...

The domain of $$f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}$$ is

Let $$f:R \to R$$ be a function such that $$f(x) = {{{x^2} + 2x + 1} \over {{x^2} + 1}}$$. Then

The number of functions $$f:\{ 1,2,3,4\} \to \{ a \in Z|a| \le 8\} $$ satisfying $$f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\} $$ is...

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 ...

Let $$f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N$$, and $$f(4) = 133,f(5) = 255$$. Then the sum of all the positive integer divisors of $$(f(3) -...

Let $$f(x)$$ be a function such that $$f(x+y)=f(x).f(y)$$ for all $$x,y\in \mathbb{N}$$. If $$f(1)=3$$ and $$\sum\limits_{k = 1}^n {f(k) = 3279} $$, t...

If $$f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}$$, then $$f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \...

$$ \text { Let } f(x)=a x^{2}+b x+c \text { be such that } f(1)=3, f(-2)=\lambda \text { and } $$ $$f(3)=4$$. If $$f(0)+f(1)+f(-2)+f(3)=14$$, then $$\...

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}$$ and $$g: \mathbf...

Let $$f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}$$ be functions defined by $$f(a)=\alpha$$, where $$\alpha$$ is the maximum of the powers of those ...

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)...

The total number of functions, $$ f:\{1,2,3,4\} \rightarrow\{1,2,3,4,5,6\} $$ such that $$f(1)+f(2)=f(3)$$, is equal to :

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...

Let f : R $$\to$$ R be defined as f (x) = x $$-$$ 1 and g : R $$-$$ {1, $$-$$1} $$\to$$ R be defined as $$g(x) = {{{x^2}} \over {{x^2} - 1}}$$. Then t...

Let $$f(x) = {{x - 1} \over {x + 1}},\,x \in R - \{ 0, - 1,1\} $$. If $${f^{n + 1}}(x) = f({f^n}(x))$$ for all n $$\in$$ N, then $${f^6}(6) + {f^7}(7)...

Let f : N $$\to$$ R be a function such that $$f(x + y) = 2f(x)f(y)$$ for natural numbers x and y. If f(1) = 2, then the value of $$\alpha$$ for which ...

Let $$f:R \to R$$ and $$g:R \to R$$ be two functions defined by $$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$$ and $$g(x) = {{1 - 2{e^{2x}}} \over {...

The range of the function, $$f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {{{3\pi } \over 4} + x} \right) + \cos \left( {{\pi \over 4} + x} \righ...

Let f : N $$\to$$ N be a function such that f(m + n) = f(m) + f(n) for every m, n$$\in$$N. If f(6) = 18, then f(2) . f(3) is equal to :

Let f : R $$\to$$ R be defined as $$f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1$$. Then, the value of $$\sum\limits_{k = 1}^{...

Consider function f : A $$\to$$ B and g : B $$\to$$ C (A, B, C $$ \subseteq $$ R) such that (gof)$$-$$1 exists, then :

Let g : N $$\to$$ N be defined asg(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...

Let $$f:R - \left\{ {{\alpha \over 6}} \right\} \to R$$ be defined by $$f(x) = {{5x + 3} \over {6x - \alpha }}$$. Then the value of $$\alpha$$ for wh...

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| -...

Let f : R $$-$$ {3} $$ \to $$ R $$-$$ {1} be defined by f(x) = $${{x - 2} \over {x - 3}}$$.Let g : R $$ \to $$ R be given as g(x) = 2x $$-$$ 3. Then, ...

The real valued function $$f(x) = {{\cos e{c^{ - 1}}x} \over {\sqrt {x - [x]} }}$$, where [x] denotes the greatest integer less than or equal to x, is...

If the functions are defined as $$f(x) = \sqrt x $$ and $$g(x) = \sqrt {1 - x} $$, then what is the common domain of the following functions :f + g, f...

The inverse of $$y = {5^{\log x}}$$ is :

The range of a$$\in$$R for which the function f(x) = (4a $$-$$ 3)(x + loge 5) + 2(a $$-$$ 7) cot$$\left( {{x \over 2}} \right)$$ sin2$$\left( {{x \ove...

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\,i...

A function f(x) is given by $$f(x) = {{{5^x}} \over {{5^x} + 5}}$$, then the sum of the series $$f\left( {{1 \over {20}}} \right) + f\left( {{2 \over ...

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 fu...

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...

Let f : R → R be defined as f (x) = 2x – 1 and g : R - {1} → R be defined as g(x) = $${{x - {1 \over 2}} \over {x - 1}}$$. Then the composition functi...

For a suitably chosen real constant a, let a function, $$f:R - \left\{ { - a} \right\} \to R$$ be defined by $$f(x) = {{a - x} \over {a + x}}$$. Furt...

If f(x + y) = f(x)f(y) and $$\sum\limits_{x = 1}^\infty {f\left( x \right)} = 2$$ , x, y $$ \in $$ N, where N is the set of all natural number, then...

Let f : R $$ \to $$ R be a function which satisfies f(x + y) = f(x) + f(y) $$\forall $$ x, y $$ \in $$ R. If f(1) = 2 and g(n) = $$\sum\limits_{k = 1}...

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} &a...

Let ƒ : (1, 3) $$ \to $$ R be a function defined by $$f(x) = {{x\left[ x \right]} \over {1 + {x^2}}}$$ , where [x] denotes the greatest integer $$ \le...

The inverse function of f(x) = $${{{8^{2x}} - {8^{ - 2x}}} \over {{8^{2x}} + {8^{ - 2x}}}}$$, x $$ \in $$ (-1, 1), is :

If g(x) = x2 + x - 1 and (goƒ) (x) = 4x2 - 10x + 5, then ƒ$$\left( {{5 \over 4}} \right)$$ is equal to:

For x $$ \in $$ (0, 3/2), let f(x) = $$\sqrt x $$ , g(x) = tan x and h(x) = $${{1 - {x^2}} \over {1 + {x^2}}}$$. If $$\phi $$ (x) = ((hof)og)(x), th...

Let f(x) = x2 , x $$ \in $$ R. For any A $$ \subseteq $$ R, define g (A) = { x $$ \in $$ R : f(x) $$ \in $$ A}. If S = [0,4], then which one of the fo...

Let f(x) = ex – x and g(x) = x2 – x, $$\forall $$ x $$ \in $$ R. Then the set of all x $$ \in $$ R, where the function h(x) = (fog) (x) is increasing,...

The domain of the definition of the function $$f(x) = {1 \over {4 - {x^2}}} + {\log _{10}}({x^3} - x)$$ is

Let $$\sum\limits_{k = 1}^{10} {f(a + k) = 16\left( {{2^{10}} - 1} \right)} $$ where the function ƒ satisfies ƒ(x + y) = ƒ(x)ƒ(y) for all natural num...

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 ƒ(x) = ax (a > 0) be written as ƒ(x) = ƒ1 (x) + ƒ2 (x), where ƒ1 (x) is an even function of ƒ2 (x) is an odd function. Then ƒ1 (x + y) + ƒ1 (x...

If $$f(x) = {\log _e}\left( {{{1 - x} \over {1 + x}}} \right)$$, $$\left| x \right| < 1$$ then $$f\left( {{{2x} \over {1 + {x^2}}}} \right)$$ is eq...

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 :

Let f : R $$ \to $$ R be defined by f(x) = $${x \over {1 + {x^2}}},x \in R$$.   Then the range of f is :

Let fk(x) = $${1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)$$ for k = 1, 2, 3, ... Then for all x $$ \in $$ R, the value of f4(x) $$-$$ f6(x...

Let N be the set of natural numbers and two functions f and g be defined as f, g : N $$ \to $$ N such that f(n) = $$\left\{ {\matrix{ {{{n + 1} \ov...

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

For $$x \in R - \left\{ {0,1} \right\}$$, Let f1(x) = $$1\over x$$, f2 (x) = 1 – x and f3 (x) = $$1 \over {1 - x}$$ be three given functions. If a ...

Let f : A $$ \to $$ B be a function defined as f(x) = $${{x - 1} \over {x - 2}},$$ Where A = R $$-$$ {2} and B = R $$-$$ {1}. Then   f   i...

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 th...

Let f(x) = 210.x + 1 and g(x)=310.x $$-$$ 1. If (fog) (x) = x, then x is equal to :

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 $$a$$, b, c $$ \in R$$. If $$f$$(x) = ax2 + bx + c is such that $$a$$ + b + c = 3 and $$f$$(x + y) = $$f$$(x) + $$f$$(y) + xy, $$\forall x,y \in R...

For x $$ \in $$ R, x $$ \ne $$ 0, Let f0(x) = $${1 \over {1 - x}}$$ and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, . . . . Then the value of f100(3) + f1$$...

If $f(x)+2 f\left(\frac{1}{x}\right)=3 x, x \neq 0$, and $\mathrm{S}=\{x \in \mathbf{R}: f(x)=f(-x)\}$; then $\mathrm{S}:$

The domain of the function f(x) = $${1 \over {\sqrt {\left| x \right| - x} }}$$ 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 \rig...

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

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 an...

The largest interval lying in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$ for which the function $$f\left( x \right) = {4^{ - {x^2}}} + {\co...

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 ont...

A real valued function f(x) satisfies the functional equation f(x - y) = f(x)f(y) - f(a - x)f(a + y) where a is given constant and f(0) = 1, f(2a - x)...

The range of the function f(x) = $${}^{7 - x}{P_{x - 3}}$$ 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

The domain of the function $$f\left( x \right) = {{{{\sin }^{ - 1}}\left( {x - 3} \right)} \over {\sqrt {9 - {x^2}} }}$$

The graph of the function y = f(x) is symmetrical about the line x = 2, then

The function $$f\left( x \right)$$ $$ = \log \left( {x + \sqrt {{x^2} + 1} } \right)$$, 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\,...

If $$f:R \to R$$ satisfies $$f$$(x + y) = $$f$$(x) + $$f$$(y), for all x, y $$ \in $$ R and $$f$$(1) = 7, then $$\sum\limits_{r = 1}^n {f\left( r \rig...

Domain of definition of the function f(x) = $${3 \over {4 - {x^2}}}$$ + $${\log _{10}}\left( {{x^3} - x} \right)$$, is

The period of $${\sin ^2}\theta $$ is

Which one is not periodic?

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

Let $$\mathrm{A}=\{1,2,3,4,5\}$$ and $$\mathrm{B}=\{1,2,3,4,5,6\}$$. Then the number of functions $$f: \mathrm{A} \rightarrow \mathrm{B}$$ satisfying ...

Let $$\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$$ and $$\mathrm{S}=\{1,2,3,4\}$$. Total number of onto functions $$f: ...

If domain of the function $$\log _{e}\left(\frac{6 x^{2}+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^{2}-3 x+4}{3 x-5}\right)$$ is $$(\alpha, \beta...

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ wi...

Let $$S=\{1,2,3,4,5,6\}$$. Then the number of one-one functions $$f: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})$$, where $$\mathrm{P}(\mathrm{S})$$...

Suppose $$f$$ is a function satisfying $$f(x + y) = f(x) + f(y)$$ for all $$x,y \in N$$ and $$f(1) = {1 \over 5}$$. If $$\sum\limits_{n = 1}^m {{{f(n)...

For some a, b, c $$\in\mathbb{N}$$, let $$f(x) = ax - 3$$ and $$\mathrm{g(x)=x^b+c,x\in\mathbb{R}}$$. If $${(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over...

For $$\mathrm{p}, \mathrm{q} \in \mathbf{R}$$, consider the real valued function $$f(x)=(x-\mathrm{p})^{2}-\mathrm{q}, x \in \mathbf{R}$$ and $$\mathr...

The number of functions $$f$$, from the set $$\mathrm{A}=\left\{x \in \mathbf{N}: x^{2}-10 x+9 \leq 0\right\}$$ to the set $$\mathrm{B}=\left\{\mathrm...

Let $$f(x)=2 x^{2}-x-1$$ and $$\mathrm{S}=\{n \in \mathbb{Z}:|f(n)| \leq 800\}$$. Then, the value of $$\sum\limits_{n \in S} f(n)$$ is equal to _____...

Let $$f(x)$$ be a quadratic polynomial with leading coefficient 1 such that $$f(0)=p, p \neq 0$$, and $$f(1)=\frac{1}{3}$$. If the equations $$f(x)=0$...

Let f(x) and g(x) be two real polynomials of degree 2 and 1 respectively. If $$f(g(x)) = 8{x^2} - 2x$$ and $$g(f(x)) = 4{x^2} + 6x + 1$$, then the val...

Let c, k $$\in$$ R. If $$f(x) = (c + 1){x^2} + (1 - {c^2})x + 2k$$ and $$f(x + y) = f(x) + f(y) - xy$$, for all x, y $$\in$$ R, then the value of $$|2...

Let S = {1, 2, 3, 4}. Then the number of elements in the set { f : S $$\times$$ S $$\to$$ S : f is onto and f (a, b) = f (b, a) $$\ge$$ a $$\forall$$ ...

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define f : S $$\to$$ S as $$f(n) = \left\{ {\matrix{ {2n} & , & {if\,n = 1,2,3,4,5} \cr {2n - 11} & ,...

Let f : R $$\to$$ R be a function defined by $$f(x) = {{2{e^{2x}}} \over {{e^{2x}} + e}}$$. Then $$f\left( {{1 \over {100}}} \right) + f\left( {{2 \ov...

Let $$f:R \to R$$ be a function defined by $$f(x) = {\left( {2\left( {1 - {{{x^{25}}} \over 2}} \right)(2 + {x^{25}})} \right)^{{1 \over {50}}}}$$. If...

The number of one-one functions f : {a, b, c, d} $$\to$$ {0, 1, 2, ......, 10} such that 2f(a) $$-$$ f(b) + 3f(c) + f(d) = 0 is ___________....

Let S = {1, 2, 3, 4, 5, 6, 7}. Then the number of possible functions f : S $$\to$$ S such that f(m . n) = f(m) . f(n) for every m, n $$\in$$ S and m ....

Let A = {0, 1, 2, 3, 4, 5, 6, 7}. Then the number of bijective functions f : A $$\to$$ A such that f(1) + f(2) = 3 $$-$$ f(3) is equal to

If f(x) and g(x) are two polynomials such that the polynomial P(x) = f(x3) + x g(x3) is divisible by x2 + x + 1, then P(1) is equal to ___________....

If a + $$\alpha$$ = 1, b + $$\beta$$ = 2 and $$af(x) + \alpha f\left( {{1 \over x}} \right) = bx + {\beta \over x},x \ne 0$$, then the value of the e...

Suppose that a function f : R $$ \to $$ R satisfies f(x + y) = f(x)f(y) for all x, y $$ \in $$ R and f(1) = 3. If $$\sum\limits_{i = 1}^n {f(i)} = 36...

Let A = {a, b, c} and B = {1, 2, 3, 4}. Then the number of elements in the set C = {f : A $$ \to $$ B | 2 $$ \in $$ f(A) and f is not one-one} is ____...