Functions

Let $f(x)=1-x, g(x)=\frac{1}{1-x}, h(x)=\frac{1}{x}$ be three functions, for $x \neq(0,1)$. If a function $F(x)$ satisfies $f(F(h(x)))=g(x)$, then

The set of all real values of $x$ for which $f(x)=\log _2\left(2^x-2\right)+\sqrt{1-x}$ is also real is

If $D$ is the domain and $G$ is the range of the real valued function $f(x)=\sqrt{\frac{1-x^2}{1+x^2}}$, then $D \cap G=$

Let $f: A \rightarrow B$ be defined as $f(x)=\frac{1}{2}-\tan \left(\frac{\pi x}{2}\right)$ and $g: B \rightarrow C$ be defined as $g(x)=\sqrt{3+4 x-4 x^2}$. If $A, B$ and $C$ are subsets of $R$ and $f$ is an onto function, then the range o...

The domain of the real valued function $f(x)=\frac{\sqrt{6 x^2+5 x-6}}{\sqrt{4-x}-\sqrt{x+4}}$ is

If $[x]$ represents the greatest integer $\leq x$, then the range of the real valued function $f(x)=\frac{1}{\sqrt{[x]^2+[x]-2}}$ is

Let $R$ be the set of all real numbers. Let $f: R \rightarrow R$ be a function defined by
$$ f(x)=\left\{\begin{array}{rcc} 2 x-5, & \text { if } & x<-3 \\ x+2, & \text { if } & -3 \leq x<5 \\ 3 x+1, & \text { if } & x \geq 5 \end{array}\ri...

Let $R$ be the set of all real number
Statement I The function $f:\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \rightarrow R$ defined by $f(x)=\sec x+\tan x$ is one-one function.
Statement II The function $f:[0, \infty) \rightarrow R$ defined...

If $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined by $f(x)=x+2|x+1|+2|x-1|$, then the element in the co-domain, which has unique pre image in the domain is

Let $H(x)=3 x^4+6 x^3-2 x^2+1$ and $g(x)$ be a polynomial of degree one. If
$\frac{H(x)}{(x-1)(x+1)(x-2)}=f(x)+\frac{g(x)}{(x-1)(x+1)(x-2)}$ then
$H(-1)+2 H(2)-3 H(1)=$

Given that for any $n \in \mathbf{N}$ there exist an odd integer $q$ and a non-negative integer $r$ such that, $n$ can be written uniquely as $n=q \times 2^r$.
Let $f: \mathbf{N} \rightarrow \mathbf{N} \times \mathbf{N}$ be function defined...

If $\frac{x^5-5}{x^3+x^2}=f(x)+\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x+1}$, then the larger value of $K$ for which $f(K)+A+B+C=1$, is

If $f: Z \rightarrow N$ is defined by
$$ f(n)=\left\{\begin{array}{cll} 2 n, & \text { if } & n>0 \\ 1, & \text { if } & n=0, \text { then } f \text { is } \\ -2 n-1, & \text { if } & n<0 \end{array}\right. $$

If $\operatorname{sech}^{-1}(1 / 2)-\operatorname{cosech}^{-1}(3 / 4)=\log _e k$, then

Let $[x]$ denote the greatest integer not more than $x$. If $A$ and $B$ are the domains of the functions $f(x)=\frac{x-[x]}{\sqrt{|x|-x}}$ and $g(x)=\frac{x-[x]}{\sqrt{|x|+x}}$ respectively, then

If $f:[-3,2] \rightarrow[0, \sqrt[3]{x}]$ is an onto function defined by $f(n)=\left\{\begin{array}{cc}2+\sqrt[3]{n}, & -3 \leq n \leq-1 \\ n^{2 / 3}, & -1 \leq n \leq 2\end{array}\right.$, then $x=$

For each $n \in \mathbf{N}$, let $A_n=\{(n+1) k / k \in \mathbf{N}\}$ and $X=\bigcup_{n \in \mathbf{N}} A_n \cdot A$ mapping $f: X \rightarrow N$ defined by $f(x)=x$, $\forall x \in \mathbf{X}$, is

The number of bijective functions $f: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $f(x+y)=f(x)+f(y) \forall x, y \in \mathbf{Z}$, is

The domain of the function, $f(x)=\sqrt{\log _{10}\left(\frac{5 x-x^2}{4}\right)}$ is

Let $f:[0,10] \rightarrow[1,20]$ be a function defined as
$$ f(x)=\left\{\begin{array}{ll} \frac{60-5 x}{3}, & 0 \leq x \leq 6 \\ 10, & 6 \leq x \leq 7 \\ 31-3 x, & 7 \leq x \leq 10 \end{array} \text { then } f\right. \text { is } $$

A function $f: \mathbf{R} \rightarrow \mathbf{R}$ is such that $f(\mathrm{l})=2$ and $f(x+y)=f(x) \cdot f(y) \forall x, y$. The area (in square units) enclosed by the lines $2|x|+5|y| \leq 4$ expressed interms of $f(1), f(2)$ and $f(4)$ is

Let $[\cdot]$ denote greatest integer function. If $f(x)=[x]$ and $g(x)=3\left[\frac{x}{3}\right]$, then the set of all real $x$ such that $f(x)=g(x)$ is

If $f(x)=x-\frac{1}{x}, x \neq 0$, then $3 f(x)=$