Subject Experts Solutions for Chapter: Relations and Functions, Exercise 1: PRACTICE EXERCISE

Let $X$ be the set of all persons living in Delhi. The persons, $a$ and $b$ in $X$ are said to be related, if the difference in their ages is at most $5$ years. The relation is

If $g\left(x\right)=\frac{1}{f\left(x\right)}$ and $f\left(x\right)=x, x\ne 0$, then which one of the following is correct?

$f\left(xy\right)=f\left(x\right)+f\left(y\right)$ is true for all

Consider the following statements: 

Statement $\mathrm{I}$: The function $f:R\to R$ such that $f\left(x\right)=x^3$ for all $x\in R$ is one-one.

Statement $\mathrm{II}$: $f\left(a\right)=f\left(b\right)\Rightarrow a=b$ for all $a, b\in R$, if the function $f$ is one-one.

Which one of the following is correct in respect of the above statements?

Suppose there is a relation $*$ between the positive numbers $x$ and $y$ given by $x*y$, if and only if $x\le y^2$. Then, which one of the following is correct?

If $f\left(x_1\right)-f\left(x_2\right)=f\left(\frac{x_1-x_2}{1-x_1{x}_2}\right)$ for $x_1, x_2\in \left(-1, 1\right)$, then what is $f\left(x\right)$ equal to $?$

What is the range of the function $y=\frac{x^2}{1+x^2},$ where $x\in R$ ?

Let $R$ be a relation on the set $N$ of natural numbers defined by $\text{'}nRm\iff n$ is a factor of $m\text{'}$. Then, which one of the following is correct?