Additional Questions - Chapter 1 Relations & Functions 10th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Additional Questions
Question 1 .
Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{-1,2,3,4,5,6,7,8,9,10,11,12\}$ Let $\mathrm{R}=\{(1,3),(2,6),(3,10),(4$, 9) $\mathcal{C} \mathrm{A} \times \mathrm{B}$ be a relation. Show that $\mathrm{R}$ is a function and find its domain, co-domain and the range of $R$.
Answer:
Domain of $R=\{1,2,3,4\}$
Co-domain of $\mathrm{R}=\mathrm{B}=\{-1,2,3,4,5,6,7,9,10,11,12\}$
Range of $\mathrm{R}=\{3,6,10,9\}$

Question 2.
Let $\mathrm{A}=\{0,1,2,3\}$ and $\mathrm{B}=\{1,3,5,7,9\}$ be two sets. Let $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ be a function given by $f(\mathrm{x})=$ $2 \mathrm{x}+1$. Represent this function as (i) a set of ordered pairs (ii) a table (iii) an arrow and (iv) $a$ graph.
Solution:
$\mathrm{A}=\{0,1,2,3\}, \mathrm{B}=\{1,3,5,7,9\}$
$f(x)=2 x+1$
$f(0)=2(0)+1=1$
$f(1)=2(1)+1=3$
$f(2)=2(2)+1=5$
$f(3)=2(3)+1=7$
(i) $\mathrm{A}$ set of ordered pairs.
$f=\{(0,1),(1,3),(2,5),(3,7)\}$
(ii) $\mathrm{A}$ table

Question 3.

State whether the graph represent a function. Use vertical line test

Solution:

It is not a function as the vertical line PQ cuts the graph at two points.

Question 4 .
Let $f=\{(2,7),(3,4),(7,9),(-1,6),(0,2),(5,3)\}$ be a function from $A=\{-1,0,2,3,5,7\}$ to $B=$ $\{2,3,4,6,7,9\}$. Is this (i) an one-one function (ii) an onto function, (iii) both one- one and onto function?
Solution:
It is both one-one and onto function.

All the elements in A have their separate images in B. All the elements in B have their preimage in A. Therefore it is one-one and onto function.
 

Question $5 .$
A function $\mathrm{f}:(-7,6) \rightarrow \mathrm{R}$ is defined as follows.
$f(x)=\left\{\begin{array}{cc}
x^{2}+2 x+1 & -7 \leq x<-5 \\
x+5 & -5 \leq x \leq 2 \\
x-1 & 2 \end{array}\right.$
Find (i) $2 f(-4)+3 f(2)$
(ii) $f(-7)-f(-3)$
Solution:
$f(x)= \begin{cases}x^{2}+2 x+1:-7 \leq x<-5 \\ x+5 & :-5 \leq x<-2 \\ x-1 & : 2 (i) $2 \mathrm{f}(-4)+3 \mathrm{f}(2)$
$f(-4)=x+5=-4+5=1$
$2 \mathrm{f}(-4)=2 \times 1=2$
$f(2)=x+5=2+5=7$
$3 \mathrm{f}(2)=3(7)=21$
$\therefore 2 \mathrm{f}(-4)+3 \mathrm{f}(2)=2+21=23$

(ii) $f(-7)=x^{2}+2 x+1$
$=(-7)^{2}+2(-7)+1$
$=49-14+1=36$
$f(3)=x+5=-3+5=2$
$f(-7)-f(-3)=36-2=34$

Question 6.
If $\mathrm{A}=\{2,3,5\}$ and $\mathrm{B}=\{1,4\}$ then find
(i) $\mathrm{A} \times \mathrm{B}$
(ii) $\mathrm{B} \times \mathrm{A}$
Answer:
$\mathrm{A}=\{2,3,5\}$
$B=\{1,4\}$
(i) $\mathrm{A} \times \mathrm{B}=\{2,3,5\} \times\{1,4\}$
$=\{(2,1)(2,4)(3,1)(3,4)(5,1)(5,4)\}$.
(ii) $\mathrm{B} \times \mathrm{A}=\{1,4\} \times\{2,3,5\}$
$=\{(1,2)(1,3)(1,5)(4,2)(4,3)(4,5)\}$

Question 7.
Let $A=\{5,6,7,8\}$;
$B=\{-11,4,7,-10,-7,-9,-13\}$ and
$f=\{(x, y): y=3-2 x, x \in A, y \in B\}$
(i) Write down the elements of $f$.
(ii) What is the co-domain?
(iii) What is the range?
(iv) Identify the type of function.
Answer:

Given, $\mathrm{A}=\{5,6,7,8\}$,
$
\begin{aligned}
& B=\{-11,4,7,-10,-7,-9,-13\} \\
& y=3-2 x
\end{aligned}
$
ie; $f(x)=3-2 x$
$
\begin{aligned}
& \mathrm{f}(5)=3-2(5)=3-10=-7 \\
& \mathrm{f}(6)=3-2(6)=3-12=-9 \\
& \mathrm{f}(7)=3-2(7)=3-14=-11 \\
& \mathrm{f}(8)=3-2(8)=3-16=-13
\end{aligned}
$

$\text { (i) } f=\{(5,-7),(6,-9),(7,-11),(8,-13)\}$

(ii) Co-domain (B)
$=\{-11,4,7,-10,-7,-9,-13\} \mathrm{i}$
(iii) Range $=\{-7,-9,-11,-13\}$
(iv) It is one-one function.

Question 8 .
A function $\mathrm{f}:[1,6] \rightarrow \mathrm{R}$ is defined as follows:
Find the value of (i) $f(5)$
(ii) $\mathrm{f}(3)$
(iii) $f(2)-f(4)$.
Solution:

$f(x)= \begin{cases}1+x & : 1 \leq x<2 \\ 2 x-1 & : 2 \leq x<4 \\ 3 x^2-10: 4 \leq x<6\end{cases}$

$\begin{aligned}
& \text { (i) } f(5)=3 x^2-10 \\
& =3\left(5^2\right)-10=75-10=65
\end{aligned}$

$\begin{aligned}
& \text { (ii) } f(3)=2 x-1 \\
& =2(3)-1=6-1=5
\end{aligned}$

$\begin{aligned}
& \text { (iii) } \mathrm{f}(2)-\mathrm{f}(4) \\
& \mathrm{f}(2)=2 \mathrm{x}-1 \\
& =2(2)-1=3 \\
& \mathrm{f}(4)=3 \mathrm{x}^2-10 \\
& =3\left(4^2\right)-10=38 \\
& \therefore \mathrm{f}(2)-\mathrm{f}(4)=3-38=35
\end{aligned}$

Question $9 .$
The following table represents a function from $\mathrm{A}=\{5,6,8,10\}$ to $\mathrm{B}=\{19,15,9,11\}$, where $f(x)$ $=2 \mathrm{x}-1$. Find the values of $\mathrm{a}$ and $\mathrm{b}$.
Solution:

$\begin{aligned}
&A=\{5,6,8,10\}, B=\{19,15,9,11\} \\
&f(x)=2 x-1 \\
&f(5)=2(5)-1=9 \\
&f(8)=2(8)-1=15 \\
&\therefore a=9, b=15
\end{aligned}$

Question 10 .
If $R=\{(a,-2),(-5,6),(8, c),(d,-1)\}$ represents the identity function, find the values of $a, b, c$ and $d$.

Solution:
$\mathrm{R}=\{(\mathrm{a},-2),(-5, \mathrm{~b}),(8, \mathrm{c}),(\mathrm{d},-1)\}$ represents the identity function.
$\mathrm{a}=-2, \mathrm{~b}=-5, \mathrm{c}=8, \mathrm{~d}=-1$