Functions

If $$f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$$, then range of \((f o g)(x)\) is

The function $f: \mathbf{N}-\{1\} \rightarrow \mathbf{N}$; defined by $f(\mathrm{n})=$ the highest prime factor of $\mathrm{n}$, is :

Let \(f: \mathbf{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathbf{R}\) and \(g: \mathbf{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathbf{R}\) be defined as \(f(x)=\frac{2 x+3}{2 x+1}\) and \(g(x)=\frac{|x|+1}{2 x+5}\). Then, the domain ...

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be defined as
$f(x)=\left\{\begin{array}{ll}\log _{\mathrm{e}} x, & x>0 \\ \mathrm{e}^{-x}, & x \leq 0\end{array}\right.$ and
$g(x)=\left\{\begin{array}...

If the domain of the function
$f(x)=\frac{\sqrt{x^2-25}}{\left(4-x^2\right)}+\log _{10}\left(x^2+2 x-15\right)$ is $(-\infty, \alpha) \cup[\beta, \infty)$, then $\alpha^2+\beta^3$ is equal to :

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 \((\alpha, \beta) \cup(\gamma, \delta]\), then $$18\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)...

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 \(f: \mathrm{R} \rightarrow \mathrm{S}\)
such that \(f(\mathrm{a}) \neq 1\), is equal to ______...

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 \(\lceil x\rceil\) denotes the smallest integer greater than or equal to \(x\). Then among the statements
(S1) :...

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 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, in the interval $[-1,2]$, is :

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

Let \(S=\{1,2,3,4,5,6\}\). Then the number of one-one functions \(f: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})\), where \(\mathrm{P}(\mathrm{S})\) denote the power set of \(\mathrm{S}\), such that \(f(n) \subset f(\mathrm{~m})\) where $...

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 $m, n \in A$ with $m \cdot n \in A$ is equal to ___________.

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

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 \(\sum\limits_{n = 1}^m {{{f(n)} \over {n(n + 1)(n + 2)}} = {1 \over {12}}}\), then \(m\) is equal to __________.

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

The domain of \(f(x) = {{{{\log }_{(x + 1)}}(x - 2)} \over {{e^{2{{\log }_e}x}} - (2x + 3)}},x \in \mathbb{R}\) 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 \(\frac{1}{f(2022)}+\frac{1}{f(2028)}\) is equal to

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 \({(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}\), then \((fog)(ac) + (gof)(b)\) is equal to ____________.

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 \((f(3) - f(2))\) 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 that the range of \(f\) is [0, 2]. Then the value of \(m\) is _________

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

If \(f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}\), then \(f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right)\) is equal to

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 \(\sum\limits_{k = 1}^n {f(k) = 3279}\), then the value of n is

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
{{{\sin }^2}x} & {{{\cos }^2}x} & {1 + \sin 2x} \cr

} } \right|,\,x \in \left[ {{\pi \ov...

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

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 \(f(0)\) is equal to

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

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 function \(\mathrm{g}: \mathrm{D} \rightarrow \mathbb{R}\) defined by $$\mathrm{g}(x)...

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 \(f(1)+f(2)=f(4)-1\) is equal to __________.

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 \(x\) )

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 \(f(f(x)) + f\left( {f\left( {{4 \over x}} \right)} \right)\) is :

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 \(|2(f(1) + f(2) + f(3) + \,\,......\,\, + \,\,f(20))|\) is equal to ____________.

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 value of \(f(2) + g(2)\) is _________.

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

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 \(\forall\) (a, b) \(\in\) S \(\times\) S } is ______________.

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 \(\mathrm{q}>0\). Let \(\mathrm{a}_{1}\), \(\mathrm{a}_{2^{\prime}}\) \(\mathrm{a}_{3}\) and $$\ma...

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 \(g: \mathbf{R} \rightarrow \mathbf{R}\) be such that \(g(f(x))=x\) for all \(x \in \mathbf{R}\). If $...

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

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 \over {100}}} \right) + f\left( {{3 \over {100}}} \right) + \,\,\,.....\,\,\, + \,\,\,f\left(...

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

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 \(f, g: \mathbb{N}-\{1\} \rightarrow \mathbb{N}\) be functions defined by \(f(a)=\alpha\), where \(\alpha\) is the maximum of the powers of those primes \(p\) such that \(p^{\alpha}\) divides \(a\), and \(g(a)=a+1\), for all $$a \in \ma...

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 \(\mathrm{B}=\left\{\mathrm{n}^{2}: \mathrm{n} \in \mathbf{N}\right\}\) such that \(f(x) \leq(x-3)^{2}+1\), for every...

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 \({f^6}(6) + {f^7}(7)\) is equal to :

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 the function fog is :

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 function \(g(x) = f(f(f(x))) + f(f(x))\), then the greatest integer less than or equa...

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 \(g(x) = {{1 - 2{e^{2x}}} \over {{e^x}}}\). Then, for which of the following range of \(\alpha\), the inequality $$f\left( ...

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
\(\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)}\)
holds, is :

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 :