Types of Functions

Let $f:R\to R$ defined by $f\left(x\right)=\left\{\begin{array}{c}\begin{array}{ll}1,& \text{if} x>0\end{array}\\ \begin{array}{ll}0,& \text{if} x=0:\end{array}\\ -1, \text{if} x<0\end{array}\right.$ then $f$ is

Let $A$ and $B$ are sets. Then $f:A\times B\to B\times A$ such that $f\left(a,b\right)=b,a$ is

Let $f:R\to R$ given by $f\left(x\right)=\left[x\right]$ where $\left[x\right]$ denote the greatest integer less than or equal to $x,$  then $f$ is

Let $f:R\to R$ given by $f\left(x\right)=\left|x\right|;$ then $f$ is

Let $A=\left\{1,2,3\right\}, B=\left\{4,5,6,7\right\}$ and $f:A\to B$ such that $f=\left\{\left(1,4\right),\left(2,5\right),\left(3,6\right)\right\}$ then $f$ is

Let $A=\left\{0,1,2,3\right\}, B=\left\{-1,1,3,5\right\}$ and $f:A\to B$ defined by $f\left(x\right)=2x-1$ then $f$ is

Let $f:N\to N$ be defined as $f\left(x\right)=2x+1.$ Then $f$ is

Let $f:N\to N$ be defined as $f\left(x\right)=3x.$ Choose the correct answer.

Let $f:R\to R$ be defined as $f\left(x\right)=3x.$ Choose the correct answer.

Let $f:R\to R$ be defined as $f\left(x\right)=x^4$, choose the correct answer.

Show that the function $f:\mathrm{R}⟶\mathrm{R}$ defined by $f\left(x\right)=\frac{x}{x^2+1}$, for all $x\in \mathrm{R}$ is neither one-one nor onto function.

Is the function $f:\mathrm{N}⟶\mathrm{N}$, where $\mathrm{N}$ is the set of natural numbers, defined by $f\left(x\right)=2x+3$ for all $x\in \mathrm{N}$ surjective?

Find whether the following function is surjective, injective or bijective:
$f:\mathrm{N}⟶\mathrm{N}$ defined by $f\left(x\right)=2x+5, x\in \mathrm{N}$

Find whether the following function is surjective, injective or bijective:
$f:\mathrm{Q}⟶\mathrm{Q}$ defined by $f\left(x\right)=3x+5, x\in \mathrm{Q}, \mathrm{Q}$ being the set of all rational numbers

Show that the function $f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=\sin x$ for $x\in \mathrm{R}$ is neither one-one nor onto function.

Show that the function $f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=\cos x$ for $x\in \mathrm{R}$ is neither one to one nor onto function.

Given, $\mathrm{A}=\{1,2,4\}, \mathrm{B}=\{3,5,7,8\}$ and $\mathrm{C}=\{3,7,8\}$. Construct an example with the figure for the following function:
A many-one onto mapping from $B$ to $A$.

Given, $\mathrm{A}=\{1,2,4\}, \mathrm{B}=\{3,5,7,8\}$ and $\mathrm{C}=\{3,7,8\}$. Construct an example with the figure for the following function:
A many-one into mapping from $A$ to $B$

Given, $\mathrm{A}=\{1,2,4\}, \mathrm{B}=\{3,5,7,8\}$ and $\mathrm{C}=\{3,7,8\}$. Construct an example with the figure for the following function:
An injective mapping (or function) from $A$ to $B$ and also from $A$ to $C$