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.

Test whether the map $f : Z\to Z$ defined by $f\left(x\right)=\left|x\right|$ is injective and/ or surjective.

Which of the following are injections, surjections or bijections? Justify your answers

$f: (0, \infty )\to R$ defined by $f\left(x\right)={\log }_{e}x$

Which of the following are injections, surjections or bijections? Justify your answers

$f: R\to (0, \infty )$ defined by $f\left(x\right)=2^x$

Which of the following are injections, surjections or bijections? Justify your answers

$f: R\to R$ defined by $f\left(x\right)=\frac{2x+1}{3}$

If $A = \left\{x\right|-1\le x\le 1\}, f(x)=x^2, g(x)=x^3,$ which of the following are surjections.

$g: A\to A$

If $A = \left\{x\right|-1\le x\le 1\}, f(x)=x^2, g(x)=x^3,$ which of the following are surjections.

$f: A\to A$

If $f(x+y)=f\left(xy\right) \forall x, y\in R$ then prove that $f$ is a constant function.

If $A = \{-2, -1, 0, 1, 2\}$ and $f: A \to B$ is a surjection defined by $f\left(x\right) = x^2+x+1$, then find $B$

Let $\mathrm{F}$ denotes the set of all onto functions from $\mathrm{A}=\left\{a_1,a_2,a_3,a_4\right\}$ to $B=\left\{x,y,z\right\}$. A function $f$ is chosen at random from $\mathrm{F}.$ The probability that $f^{–1}\left(\mathrm{t}\right)$ consists of exactly one element is

Let $R$ be the set of all real numbers and $f:R\to R$ be a continuous function. Suppose $\left|f\right(x)-f(y\left)\right|\ge |x-y|$ for all real numbers $x$ and $y.$ Then

A bijective function is both one-one and onto.

Let $f:\left[0,1\right]\to R$ be an injective continuous function that satisifes the condition $-1<f\left(0\right)<f\left(1\right)<1$ Then, the number of functions $g:\left[-1,1\right]\to \left[0,1\right]$ such that $\left(gof\right)\left(x\right)=x$ for all $x\in \left[0,1\right]$ is

Let $f:\left[0,1\right]\to \left[-1,1\right]$ and $g:\left[-1,1\right]\to \left[0,2\right]$ be two functions such that $g$ is injective and $gof:\left[0,1\right]\to \left[0,2\right]$ is surjective. Then,

Let $f:R\to R$ be a continuous function such that $f\left(x^2\right)=f\left(x^3\right)$ for all $x\in R$. Consider the following statements.

I. $f$ is an odd function.

II. $f$ is an even function.

III. $f$ is differentiable everywhere.

Then,

The function $f:N\to N$ given by $f\left(x\right)=2x$ is

 Let $A=\left\{x:-\frac{\pi }{2}\le x\le \frac{\pi }{2}\right\}$ and $B=\{x:-1\le x\le 1\}$. If $f:A\to B$ is defined by $f\left(x\right)=\sin x$, then $f\left(x\right)$ is