Functions

Consider the function $f:\left[\frac{1}{2},1\right]\rightarrow\,R$ defined by $f\left(x\right)=4\sqrt{2}x^{3}-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f\left(x\right)$ intersects the $x$-axis exactly at…

The number of elements in the set$\left{x\in\,R:\left(|x|-3\right)\left|x+4\right|=6\right}$ is equal to

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

If the function$f:\left(\left-\infty,-1\right]\right\rightarrow\,\left\left(a,b\right)\right]$defined by$f\left(x\right)=e^{x^{3}-3x+1}$is one-one and onto, then thedistance of the point$P\left(2b+4,a+2\right)$from the…

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

The number of bijective function $f\left(1,3,5,7,\cdots\,,99\right)\rightarrow\,\left(2,4,6,8,\cdots\,,100\right)$ if $f\left(3\right)>f\left(5\right)>f\left(7\right)\cdots\,>f\left(99\right)$ is

Let $f:R\rightarrow\,R$ be a function such that $f\left(x\right)=\frac{x^{2}+2x+1}{x^{2}+1}$. Then

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

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

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

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

Let $A=\left{1,2,3,\ldots\,,10\right}$ and $f:A\rightarrow\,A$ be defined as $f\left(k\right)=\left{\begin{matrix} k+1 & \mathrm{if}k\mathrm{is}\operatorname{odd}\, \\ k & \mathrm{if}k\mathrm{is}\operatorname{even}\,…