G Tewani Solutions for Chapter: Functions, Exercise 3: DPP

Let $f:\left(-\infty ,0\right]\to \left[1,\infty \right)$ be defined as $f\left(x\right)=\left(1+\sqrt{-x}\right)-\left(\sqrt{-x}+x\right),$ then $f\left(x\right)$ is

Let $A$ be a non-empty set of real numbers and $f:A\to A$ be such that $f\left(f\left(x\right)\right)=x,\forall x\in R$ Then $f\left(x\right)$ is:

$f:R\to R$ defined by $f\left(x\right)=\frac{1}{2}x\left|x\right|+\cos x+1$ is:

Let $f:R\to R,$ $f\left(x\right)=\left\{\begin{array}{ll}2x+{\alpha }^2& x\ge 2\\ \frac{\alpha x}{2}+10& x<2\end{array}\right.$ is onto function, then $\alpha$ belongs to:

Let $f:R\to \left(0,\frac{2\pi }{3}\right]$ defined as $f\left(x\right)={\cot }^{-1}\left(x^2-4x+\alpha \right)$. Then the smallest integral value of $\alpha$ such that $f\left(x\right)$ is into function, is:

Let $f:R^{+}\to \left\{-1,0,1\right\}$ be defined by $f\left(x\right)=sgn\left(x-x^4+x^7-x^8-1\right)$ where $\mathrm{sgn}$ denotes signum function. Then $f\left(x\right)$ is.

Which of the following statements are incorrect?

   $\mathrm{I}$. If $f\left(x\right)$ and $g\left(x\right)$ are one-one, then $f\left(x\right)+g\left(x\right)$ is also one-one.

   $\mathrm{II}$. If $f\left(x\right)$ and $g\left(x\right)$ are one-one, then $f\left(x\right)·g\left(x\right)$ is also one-one.

   $\mathrm{III}$. If $f\left(x\right)$ is odd, then it is necessarily one-one.

Which of the following functions is one-one?