Relations And Functions XII — CBSE Class 12 board PYQs

A function $f:R\to R$ defined as $f(x)=x^2-4x+5$ is

A relation $R$ in a set $A$ is called ____, if $(a_1,a_2)\in R$ implies $(a_2,a_1)\in R$, for all $a_1,a_2\in A$.

If $R=\{(x,y):x+2y=8\}$ is a relation on $N$, write the range of $R$.

State the reason for the relation $R$ in the set $\{1,2,3\}$ given by $R=\{(1,2),(2,1)\}$ not to be transitive.

Let $A=\{1,2,3\}$, $B=\{4,5,6,7\}$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B$. State whether $f$ is one-one or not.

What is the range of the function $f(x)=\frac{|x-1|}{(x-1)}$?

If $f(x)=x+7$ and $g(x)=x-7$, $x\in R$, find $(fog)(7)$.

If $f(x)$ is an invertible function, find the inverse of $f(x)=\frac{3x-2}{5}$.

If $f:R\to R$ is given by $f(x)={\left(3-x^3\right)}^{1/3}$, then find the value of $fof(x)$.

Consider a bijective function $f:R_{+}\to (7,\infty )$ given by $f(x)=16x^2+24x+7$, where $R_{+}$ is the set of all positive real numbers. Find the inverse function of $f$.

Students of a school are taken to a railway museum to learn about railways heritage and its history. An exhibit in the museum depicted many rail lines on the track near the railway station. Let $L$ be the set of all rail lines on the railway track and $R$ be the relation on $L$ d …

Let $S$ be the relation defined by $S=\{({\ell }_1,{\ell }_2):{\ell }_1\text{ is perpendicular to }{\ell }_2\}$. Check whether the relation $S$ is symmetric and transitive.

A function $f:[-4,4]\to [0,4]$ is given by $f(x)=\sqrt{16-x^2}$. Show that $f$ is an onto function but not a one-one function. Further, find all possible values of $\alpha$ for which $f(\alpha )=\sqrt{7}$.

Let $N$ be the set of natural numbers and $R$ be the relation on $N\times N$ defined by $(a,b)R(c,d)$ iff $ad=bc$ for all $a,b,c,d\in N$. Show that $R$ is an equivalence relation.

Check whether the relation $R$ defined on the set $A=\{1,2,3,4,5,6\}$ as $R=\{(a,b):b=a+1\}$ is reflexive, symmetric or transitive.

Let $f:N\to Y$ be a function defined as $f(x)=4x+3$, where $Y=\{y\in N:y=4x+3\text{ for some }x\in N\}$. Show that $f$ is invertible. Find its inverse.

Let $A=\{x\in Z:0\le x\le 12\}$. Show that $R=\{(a,b):a,b\in A,|a-b|\text{ is divisible by }4\}$ is an equivalence relation. Find the set of all elements related to $1$. Also write the equivalence class $[2]$.

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

Consider $f:R_{+}\to [-5,\infty )$ given by $f(x)=9x^2+6x-5$. Show that $f$ is invertible with $f^{-1}(y)=\frac{\sqrt{y+6}-1}{3}$. Hence find
(i) $f^{-1}(10)$ and
(ii) $y$ if $f^{-1}(y)=\frac{4}{3}$. (Here $R_{+}$ is the set of all non-negative real numbers.)

Let $f:N\to N$ be a function defined as $f(x)=9x^2+6x-5$. Show that $f:N\to S$, where $S$ is the range of $f$, is invertible. Find the inverse of $f$ and hence find $f^{-1}(43)$ and $f^{-1}(163)$.

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N\times N$ defined by $(a,b)R(c,d)$ if $ad(b+c)=bc(a+d)$. Show that $R$ is an equivalence relation.

If the function $f:R\to R$ is given by $f(x)=x^2+2$ and $g:R\to R$ is given by $g(x)=\frac{x}{x-1}$, $x\ne 1$, find $fog$ and $gof$, and hence find $fog(2)$ and $gof(-3)$.

Show that the function $f$ in $A=R-\left\{\frac{2}{3}\right\}$ defined as $f(x)=\frac{4x+3}{6x-4}$ is one-one and onto. Hence find $f^{-1}$.

Let $A=R-\{3\}$ and $B=R-\{1\}$. Consider the function $f:A\to B$ defined by $f(x)=\frac{x-2}{x-3}$. Show that $f$ is one-one and onto; hence find $f^{-1}$.

Let $f:R\to R$ be defined as $f(x)=10x+7$. Find the function $g:R\to R$ such that $gof=fog=I_R$.

Let $Z$ be the set of all integers and $R$ be the relation on $Z$ defined as $R=\{(a,b):a,b\in Z\text{ and }(a-b)\text{ is divisible by }5\}$. Prove that $R$ is an equivalence relation.

Show that the relation $S$ in the set $R$ of real numbers, defined as $S=\{(a,b):a,b\in R\text{ and }a\le b^3\}$, is neither reflexive, nor symmetric, nor transitive.

Prove that the relation $R$ in the set $A=\{1,2,3,4,5\}$ given by $R=\{(a,b):|a-b|\text{ is even}\}$ is an equivalence relation.

Let $T$ be the set of all triangles in a plane with $R$ as the relation in $T$ given by $R=\{(T_1,T_2):T_1\cong T_2\}$. Show that $R$ is an equivalence relation.

If $f$ is the greatest integer function and $g$ is the absolute value function, find the value of $(fog)\left(\frac{-3}{2}\right)+(gof)\left(\frac{4}{3}\right)$.

Find the inverse of the function $f(x)=4x-7$, $x\in R$.

Let $f:X\to Y$ and $g:Y\to Z$ be two invertible functions. Then prove that $gof$ is also invertible with $(gof{)}^{-1}=f^{-1}o{g}^{-1}$.