Embibe Experts Solutions for Chapter: Relations and Functions, Exercise 1: Exercise

If $f : Z\to Z$ defined by $f\left(x\right)=\left|x\right|$, then $f\left(x\right)$ is

Consider the function $f\left(x\right)=\frac{1}{x+2}$, then $f\left(x\right)$ is 

The function $f:A\to B$ defined by $f\left(x\right)=-x^2+6x-8$ is a bijective function, if

Show that the relation $R$ on $R$ is defined as $R=\left\{\right(a,b):a\le b\}$, is reflexive and transitive but not symmetric.

Prove that the following sets are equivalent:

$\left\{1,2,3,4,5,6,...\right\}$

$\left\{2,4,6,8,10,...\right\}$

$\left\{1,3,5,7,9,...\right\}$

$\left\{1,4,9,16,25,...\right\}$

Test whether the map $f:\mathrm{N}\to \mathrm{N}$, defined by $f\left(x\right)=2x+4$ is injective and/ or surjective.

The diagram shows the graph of a function $f$ from $A$ to $B$.

State whether $f$ is a one-to-one function, giving a reason for your answer.

Show that the function $f:R\to R$ defined by $f\left(x\right)=\frac{x}{x^2+1},\forall x\in \mathrm{ℝ}$ is neither one-one nor onto. Also, if $g:R\to R$ is defined as $g\left(x\right)=2 x-1$, find $fog\left(\mathrm{x}\right)$.