Types of Functions

Let $f=\left\{\right(2,7),(3,4),(7,9),(-1,6),(0,2),(5,3\left)\right\}$ be a function from  $A=\{-1,0,2,3,5,7\}$ to $B=\{2,3,4,6,7,9\}$,find wether it is one-one and onto function or not

Let $f:R\to R$ and $f\left(x\right)=5x-7$, show that it is bijective.

Let $f:N\to Q$ and $f\left(x\right)=x^3$,find it is bijective or not.

Show that the function $f:\mathrm{R}\to \{y\in \mathrm{R}:-1<\mathrm{y}<1\}$ defined by $f\left(x\right)=\frac{x}{1+\left|x\right|},x\in \mathrm{R}$ is one-one and onto function.

Suppose $f_1$ and $f_2$ are non-zero one-one functions from $\mathrm{R}$ to $\mathrm{R}$. Is $\frac{f_1}{f_2}$ necessarily one-one? Justify your answer. Here$\frac{f_1}{f_2}:\mathrm{R}\to \mathrm{R}$ is given by  $\left(\frac{f_1}{f_2}\right)\left(x\right)=\frac{{\displaystyle f_1\left(x\right)}}{{\displaystyle f_2\left(x\right)}}, x\in \mathrm{R}$.

Show that if $f_1$ and $f_2$ are one-one maps from $\mathrm{R}$ to $\mathrm{R}$ then the product $f_1·f_2:\mathrm{R}\to \mathrm{R}$ defined by $(f_1·f_2)\left(x\right)=f_1\left(x\right)·f_2\left(x\right)$ need not be one-one.

Show that the function $f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=ax+b$, where $a,b\in \mathrm{R}, \mathrm{a}\ne 0$ is a bijection.

Show that the function $f:\mathrm{N}\to \mathrm{N}$ defined by $f\left(n\right)=n-(-1{)}^n$ is bijective.

Let $\mathrm{A}=\mathrm{R}-\left\{1\right\}$and $\mathrm{B}=-\left\{1\right\}.$ consider the function $f:\mathrm{A}\to \mathrm{B}$ defined by $f\left(x\right)=\frac{x+1}{x-1}$, Is  $f$ one-one and onto$?$ justify your answer.

In the following case, state whether the function is injective, surjective or bijective justify your answer;

$f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=2x^3-7$

In the following case, state whether the function is injective, surjective or bijective justify your answer;

$f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=4x^3+7$

In the following case, state whether the function is injective, surjective or bijective justify your answer;

$f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=x+\left|x\right|$

In the following case, state whether the function is injective, surjective or bijective justify your answer;

$f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=1+x^2$

In the following case, state whether the function is injective, surjective or bijective justify your answer;

$f:\mathrm{R}\to \mathrm{R}$ defined by $f\left(x\right)=3-4x$

Show that the function $f:\mathrm{R}\to \mathrm{R},$ Given by 

$f\left(x\right)=\left\{\begin{array}{l}1 , \mathrm{if} x \mathrm{is} \mathrm{rational} \\ -1 , \mathrm{if} x \mathrm{is} \mathrm{irrational}\end{array}\right.$

is neither one-one nor onto.

Show that the Function $f: \mathrm{R}\to \mathrm{R}$, given by $f\left(x\right)=\cos x$ is neither one-one nor onto.

Show that the function $f:\mathrm{N}\to \mathrm{N}$, defined by $f\left(n\right)=n^2+n+1$ is one-one but not onto.

Let $A=B=R-\left\{\frac{2}{5}\right\}.$ Consider the function $f:A\to B$ defined by $f\left(x\right)=\frac{2x-3}{5x-2},$ show that $f$ is bijective.

Let the function $f:N-\left\{1\right\}\to N$ be defined by $f\left(n\right)=$the highest prime factor of $n.$ Show that $f$ is neither one-one nor onto.

If $f:\mathrm{X}\to \mathrm{Y}$ be defined by $f\left(x\right)=\sqrt{3}\sin x+\cos x+4$ is one-one and onto, then $\mathrm{Y}=$