Types of Functions

Let $A=R-\left\{3\right\}$ and $B=R-\left\{1\right\}.$ Consider the function $f:A\to B$ defined by $f\left(x\right)=\left(\frac{x-2}{x-3}\right).$ Is $f$ one-one and onto$?$ Justify your answer.

Let $f:R\to R$ be a continuous and one-one function. Then which of the following is true?

Which of the following graphs represents one-one function, explain with reason.

Find the complete set of values of $x$ for which ${\log }_{\sqrt{x}+1}\frac{x-1}{x-2}$ is defined

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

Given that $f:Z\to Z, f\left(x\right)=\left|x\right|$. Check whether is $f\left(x\right)$ is one-one or onto.

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.

Is ${\cos }^{-1}x$ even function or odd function?

If $gof$ is onto and $g$ is one one then show that $f$ is onto function.

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

Consider a function $f:\left[0, \frac{\pi }{2}\right]\to R$ given by $f\left(x\right)=\sin x$ and $g:\left[0, \frac{\pi }{2}\right]\to R$ given by $g\left(x\right)=\cos x$. Show that $f$ and $g$ are one-one, but $f+g$ is not one-one.

Consider the identity function $I_{\mathit{N}}:\mathit{N}\to \mathit{N}$ defined as $I_{\mathit{N}}\left(x\right)=x\forall x\in \mathit{N}$. Show that although $I_{\mathit{N}}$ is onto but $I_{\mathit{N}}+I_{\mathit{N}}:\mathit{N}\to \mathit{N}$ defined as
$\left(I_{\mathit{N}}+I_{\mathit{N}}\right)\left(x\right)=I_{\mathit{N}}\left(x\right)+I_{\mathit{N}}\left(\mathrm{x}\right)=x+x=2x$ is not onto.
 

Show that a one-one function $f:\left\{1, 2, 3\right\}\to \left\{1, 2, 3\right\}$ must be onto.

Show that an onto function $f:\left\{1, 2, 3\right\}\to \left\{1, 2, 3\right\}$ is always one-one.

Show that $f:N\to N$, given by
$f\left(x\right){=}_{x-1, \mathrm{if} x \mathrm{is} \mathrm{even}}^{x+1, \mathrm{if} x \mathrm{is} \mathrm{odd}}$

is both one-one and onto.

Show that the function $f:R\to R$, defined as $f\left(x\right)=x^2$, is neither one-one nor onto.

Show that the function $f:N\to N$, given by $f\left(1\right)=f\left(2\right)=1$ and $f\left(x\right)=x-1$, for every $x>2$, is onto but not one-one.

Prove that the function $f:R\to R$, given by $f\left(x\right)=2x$, is one-one and onto.

Show that the function $f\mathit{:}N\to N$, given by $f\left(x\right)=2x$, is one-one but not onto.