Samacheer Kalvi 10th Maths Exercise 1.3 Solutions

10th Maths Exercise 1.3 solutions (Chapter 1 Relations and Functions) with step-by-step answers for all 25 questions, as per the 2026-27 Samacheer Kalvi textbook.

Answers at a glance25 answers · full steps below

Q1Domain $=N$ , Co-domain $=N$ , Range $=\{2,4,6,\ldots\}$ (even natural numbers); yes, $f$ is a function.
Q2 $R=\{(3,10),(4,17),(6,37),(8,65)\}$ is a function from $X$ to $N$ .
Q3 (i) $f(-1)=12$
Q3 (ii) $f(2a)=4a^2-10a+6$
Q3 (iii) $f(2)=0$
Q3 (iv) $f(x-1)=x^2-7x+12$
Q4 (i)(a) $f(0)=9$
Q4 (i)(b) $f(7)=6$
Q4 (i)(c) $f(2)=6$
Q4 (i)(d) $f(10)=0$
Q4 (ii) $x=9.5$
Q4 (iii)Domain $=\{x \mid 0\le x\le 10\}$ ; Range $=\{y \mid 0\le y\le 9\}$
Q4 (iv)The image of $6$ is $5$ , that is $f(6)=5$ .
Q5 $2$
Q6 (i) $-2$
Q6 (ii) $x=\dfrac{3}{2}$
Q6 (iii) $x=3$
Q6 (iv) $x=\dfrac{1}{2}$
Q7 $V(x)=x(24-2x)^2=4x^3-96x^2+576x$ cu.cm
Q8 $x=1$
Q9 $d(t)=500t$ km, $t\ge 0$
Q10 (i)Yes, the relation $y=ax+b$ is a function.
Q10 (ii) $a=0.9,\ b=24.5$ so $y=0.9x+24.5$
Q10 (iii) $60.5$ inches
Q10 (iv) $32$ cm

Q1. Let $f = \{(x,y) \mid x, y \in N$ and $y = 2x\}$ be a relation on N. Find the domain …

Let $f = \{(x,y) \mid x, y \in N$ and $y = 2x\}$ be a relation on N. Find the domain, co-domain and range. Is this relation a function?

Show steps (6)

$y=2x$ with $x,y\in N$ gives the pairs $(1,2),(2,4),(3,6),(4,8),\ldots$
Every natural number $x$ has one and only one value $y=2x$ .
So each element of the domain has a unique image.
Domain $=N=\{1,2,3,\ldots\}$
Co-domain $=N$
Range $=\{2,4,6,8,\ldots\}$ , the set of all even natural numbers.

Answer

Domain $=N$ , Co-domain $=N$ , Range $=\{2,4,6,\ldots\}$ (even natural numbers); yes, $f$ is a function.

Q2. Let $X = \{3, 4, 6, 8\}$ . Determine whether the relation $R = \{(x, f(x)) \mid x \in X, f(x) = x^2 + 1\}$ is a function from …

Let $X = \{3, 4, 6, 8\}$ . Determine whether the relation $R = \{(x, f(x)) \mid x \in X, f(x) = x^2 + 1\}$ is a function from $X$ to $N$ ?

Show steps (7)

$f(x)=x^2+1$ for $x\in X=\{3,4,6,8\}$
$f(3)=3^2+1=10$
$f(4)=4^2+1=17$
$f(6)=6^2+1=37$
$f(8)=8^2+1=65$
$R=\{(3,10),(4,17),(6,37),(8,65)\}$
Each element of $X$ has exactly one image and all images $10,17,37,65$ belong to $N$ .

Answer

$R=\{(3,10),(4,17),(6,37),(8,65)\}$ is a function from $X$ to $N$ .

Q3. Given the function $f : x \rightarrow x^2 - 5x + 6$ , evaluate

Given the function $f : x \rightarrow x^2 - 5x + 6$ , evaluate

(i)

$f(-1)$

$f(x)=x^2-5x+6$
Put $x=-1$ :
$f(-1)=(-1)^2-5(-1)+6$
$=1+5+6=12$

Answer

$f(-1)=12$

(ii)

$f(2a)$

Put $x=2a$ :
$f(2a)=(2a)^2-5(2a)+6$
$=4a^2-10a+6$

Answer

$f(2a)=4a^2-10a+6$

(iii)

$f(2)$

Put $x=2$ :
$f(2)=2^2-5(2)+6$
$=4-10+6=0$

Answer

$f(2)=0$

(iv)

$f\left(x - 1\right)$

Put $x=x-1$ :
$f(x-1)=(x-1)^2-5(x-1)+6$
$=x^2-2x+1-5x+5+6$
$=x^2-7x+12$

Answer

$f(x-1)=x^2-7x+12$

Q4. A graph representing the function $f(x)$ is given in Fig.1.16 it is …

A graph representing the function $f(x)$ is given in Fig.1.16 it is clear that $f(9) = 2$ .

(i)(a)

Find the following values of the function (a) $f(0)$

Fig. 1.16: a graph of y = f(x) on axes numbered 1 to 10, a polyline falling from about (0,9) to a minimum near (4,3), rising to a peak near (7,6), then falling through (9,2) to about (10,0).

From the graph, the curve meets the $y$ -axis at $9$ .
So when $x=0$ , $y=9$ .

Answer

$f(0)=9$

(i)(b)

Find the following values of the function (b) $f(7)$

Draw a vertical line at $x=7$ to meet the curve.
The corresponding $y$ -value is $6$ .

Answer

$f(7)=6$

(i)(c)

Find the following values of the function (c) $f(2)$

Draw a vertical line at $x=2$ to meet the curve.
The corresponding $y$ -value is $6$ .

Answer

$f(2)=6$

(i)(d)

Find the following values of the function (d) $f(10)$

At $x=10$ the curve touches the $x$ -axis.
So the $y$ -value is $0$ .

Answer

$f(10)=0$

(ii)

For what value of $x$ is $f(x) = 1$ ?

Draw the horizontal line $y=1$ .
It cuts the curve at the point whose $x$ -value is $9.5$ .

Answer

$x=9.5$

(iii)

Describe the following (i) Domain (ii) Range.

The graph is drawn for $x$ from $0$ to $10$ .
So domain $=\{x \mid 0\le x\le 10,\ x\in R\}$ .
The $y$ -values on the curve run from the lowest $0$ to the highest $9$ .
So range $=\{y \mid 0\le y\le 9,\ y\in R\}$ .

Answer

Domain $=\{x \mid 0\le x\le 10\}$ ; Range $=\{y \mid 0\le y\le 9\}$

(iv)

What is the image of 6 under $f$ ?

Draw a vertical line at $x=6$ to meet the curve.
The corresponding $y$ -value is $5$ .

Answer

The image of $6$ is $5$ , that is $f(6)=5$ .

Q5. Let $f(x) = 2x + 5$ . If $x \neq 0$ then find $\dfrac{f(x+2) - f(2)}{x}$ .

$f(x)=2x+5$
$f(x+2)=2(x+2)+5=2x+9$
$f(2)=2(2)+5=9$
$f(x+2)-f(2)=2x+9-9=2x$
$\dfrac{f(x+2)-f(2)}{x}=\dfrac{2x}{x}=2$ (since $x\neq 0$ )

Answer

$2$

Q6. A function $f$ is defined by $f(x) = 2x - 3$ .

A function $f$ is defined by $f(x) = 2x - 3$ .

(i)

find $\dfrac{f(0) + f(1)}{2}$ .

$f(x)=2x-3$
$f(0)=2(0)-3=-3$
$f(1)=2(1)-3=-1$
$\dfrac{f(0)+f(1)}{2}=\dfrac{-3+(-1)}{2}=\dfrac{-4}{2}=-2$

Answer

$-2$

(ii)

find $x$ such that $f(x) = 0$ .

$f(x)=0$
$2x-3=0$
$2x=3$
$x=\dfrac{3}{2}$

Answer

$x=\dfrac{3}{2}$

(iii)

find $x$ such that $f(x) = x$ .

$f(x)=x$
$2x-3=x$
$2x-x=3$
$x=3$

Answer

$x=3$

(iv)

find $x$ such that $f(x) = f(1 - x)$ .

$f(1-x)=2(1-x)-3=2-2x-3=-2x-1$
$f(x)=f(1-x)$ gives $2x-3=-2x-1$
$2x+2x=-1+3$
$4x=2$
$x=\dfrac{1}{2}$

Answer

$x=\dfrac{1}{2}$

Q7. An open box is to be made from a square piece of …

An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides as shown (Fig.1.17). Express the volume $V$ of the box as a function of $x$ .

Fig. 1.17: a square sheet of side 24 cm with squares of side x cut from each corner, with bottom edge marked x, 24−2x, x, and below it the resulting open box with height x and base edge 24−2x.

Show steps (6)

Squares of side $x$ cm are cut from each corner of a square of side 24 cm.
Length of the base $=24-2x$ , breadth of the base $=24-2x$
Height of the box $=x$
$V=\text{length}\times\text{breadth}\times\text{height}$
$V=(24-2x)(24-2x)\times x$
$V(x)=x(24-2x)^2$ cubic cm, where $0<x<12$

Answer

$V(x)=x(24-2x)^2=4x^3-96x^2+576x$ cu.cm

Q8. A function $f$ is defined by $f(x) = 3 - 2x$ . Find $x$ such that …

A function $f$ is defined by $f(x) = 3 - 2x$ . Find $x$ such that $f(x^2) = (f(x))^2$ .

Show steps (9)

$f(x)=3-2x$
$f(x^2)=3-2x^2$
$(f(x))^2=(3-2x)^2=9-12x+4x^2$
Given $f(x^2)=(f(x))^2$
$3-2x^2=9-12x+4x^2$
$0=6-12x+6x^2$
$6(x^2-2x+1)=0$
$(x-1)^2=0$
$x=1$

Answer

$x=1$

Q9. A plane is flying at a speed of 500 km per hour. …

A plane is flying at a speed of 500 km per hour. Express the distance ' $d$ ' travelled by the plane as function of time $t$ in hours.

Distance $=$ speed $\times$ time
Speed $=500$ km/hr, time $=t$ hours
$d(t)=500t$
For each value of $t$ there is only one value of $d$ , so $d$ is a function of $t$ .

Answer

$d(t)=500t$ km, $t\ge 0$

Q10. The data in the adjacent table depicts the length of a person …

The data in the adjacent table depicts the length of a person forehand and their corresponding height. Based on this data, a student finds a relationship between the height ( $y$ ) and the forehand length( $x$ ) as $y = ax + b$ , where $a$ , $b$ are constants.

(i)

Check if this relation is a function.

Length ' $x$ ' of forehand (in cm)	Height ' $y$ ' (in inches)
35	56
45	65
50	69.5
55	74

Each forehand length $x$ in the table has exactly one height $y$ .
The pairs are $(35,56),(45,65),(50,69.5),(55,74)$ .
No $x$ -value is repeated with two different $y$ -values.
So the relation is a function.

Answer

Yes, the relation $y=ax+b$ is a function.

(ii)

Find $a$ and $b$ .

Length ' $x$ ' of forehand (in cm)	Height ' $y$ ' (in inches)
35	56
45	65
50	69.5
55	74

Show steps (6)

Using $(35,56)$ : $35a+b=56$ ... (1)
Using $(45,65)$ : $45a+b=65$ ... (2)
(2) $-$ (1): $10a=9$
$a=0.9$
From (1): $b=56-35(0.9)=56-31.5=24.5$
Verify with $(50,69.5)$ : $0.9(50)+24.5=69.5$ ✓

Answer

$a=0.9,\ b=24.5$ so $y=0.9x+24.5$

(iii)

Find the height of a person whose forehand length is 40 cm.

Length ' $x$ ' of forehand (in cm)	Height ' $y$ ' (in inches)
35	56
45	65
50	69.5
55	74

$y=0.9x+24.5$
Put $x=40$ :
$y=0.9(40)+24.5$
$=36+24.5=60.5$

Answer

$60.5$ inches

(iv)

Find the length of forehand of a person if the height is 53.3 inches.

Length ' $x$ ' of forehand (in cm)	Height ' $y$ ' (in inches)
35	56
45	65
50	69.5
55	74

$y=0.9x+24.5$
Put $y=53.3$ :
$53.3=0.9x+24.5$
$0.9x=53.3-24.5=28.8$
$x=\dfrac{28.8}{0.9}=32$

Answer

$32$ cm

Quick check

Let $f$ and $g$ be two functions given by $f = \{(0,1),(2,0),(3,-4),(4,2),(5,7)\}$ , $g = \{(0,2),(1,0),(2,4),(-4,2),(7,0)\}$ then the range of $f \circ g$ is
1. (1) $\{0,2,3,4,5\}$
2. (2) $\{-4,1,0,2,7\}$
3. (3) $\{1,2,3,4,5\}$
4. (4) $\{0,1,2\}$
Show answer
Answer: (4) $\{0,1,2\}$
1. $(f\circ g)(0)=f(2)=0$ , $(f\circ g)(1)=f(0)=1$ .
2. $(f\circ g)(2)=f(4)=2$ , $(f\circ g)(-4)=f(2)=0$ , $(f\circ g)(7)=f(0)=1$ .
3. Range $=\{0,1,2\}$ .
Question 1 of 5
If there are 1024 relations from a set $A = \{1, 2, 3, 4, 5\}$ to a set $B$ , then the number of elements in $B$ is
1. (1) 3
2. (2) 2
3. (3) 4
4. (4) 8
Show answer
Answer: (2) 2
1. Number of relations from $A$ to $B$ is $2^{n(A)\cdot n(B)}$ .
2. $2^{5\,n(B)}=1024=2^{10}$ .
3. $5\,n(B)=10\Rightarrow n(B)=2$ .
Question 2 of 5
Let $A = \{1,2,3,4\}$ and $B = \{4,8,9,10\}$ . A function $f : A \to B$ given by $f = \{(1,4),(2,8),(3,9),(4,10)\}$ is a
1. (1) Many-one function
2. (2) Identity function
3. (3) One-to-one function
4. (4) Into function
Show answer
Answer: (3) One-to-one function
1. Images are $4,8,9,10$ — all different.
2. Different elements of $A$ have different images.
3. So $f$ is one-to-one.
Question 3 of 5
If $f(x) = 2x^2$ and $g(x) = \dfrac{1}{3x}$ , then $f \circ g$ is
1. (1) $\dfrac{3}{2x^2}$
2. (2) $\dfrac{2}{3x^2}$
3. (3) $\dfrac{2}{9x^2}$
4. (4) $\dfrac{1}{6x^2}$
Show answer
Answer: (3) $\dfrac{2}{9x^2}$
1. $(f\circ g)(x)=f\big(g(x)\big)=f\left(\dfrac{1}{3x}\right)$ .
2. $=2\left(\dfrac{1}{3x}\right)^{2}$ .
3. $=\dfrac{2}{9x^{2}}$ .
Question 4 of 5
$A = \{a,b,p\}$ , $B = \{2,3\}$ , $C = \{p,q,r,s\}$ then $n[(A \cup C) \times B]$ is
1. (1) 8
2. (2) 20
3. (3) 12
4. (4) 16
Show answer
Answer: (3) 12
1. $A\cup C=\{a,b,p,q,r,s\}$ , so $n(A\cup C)=6$ .
2. $n(B)=2$ .
3. $n[(A\cup C)\times B]=6\times 2=12$ .
Question 5 of 5

Chapter 1 formulas

Cartesian product of A with B
$A \times B = \{(a,b) \mid \text{for all } a \in A,\ b \in B\}$
A relation R from A to B is a subset of A × B
$R \subseteq A \times B$
Condition for a relation R from X to Y to be a function
$\text{for every } x \in X \text{ there exists only one } y \in Y$
Identity function
$f(x) = x$
Reciprocal function
$f(x) = \dfrac{1}{x}$
Constant function
$f(x) = c$
Linear function
$f(x) = ax + b,\ a \neq 0$
Quadratic function
$f(x) = ax^{2} + bx + c,\ a \neq 0$
Cubic function
$f(x) = ax^{3} + bx^{2} + cx + d,\ a \neq 0$
Composition of two functions f : A → B and g : B → C
$g \circ f : A \to C,\quad (g \circ f)(x) = g(f(x)) \ \text{ for all } x \in A$
Composition is not commutative in general
$f \circ g \neq g \circ f$
Composition is associative
$f \circ (g \circ h) = (f \circ g) \circ h$

Answers checked against the Samacheer Kalvi textbook · Updated 28 September 2026

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Q1. Let f={(x,y)∣x,y∈Nf = \{(x,y) \mid x, y \in Nf={(x,y)∣x,y∈N and y=2x}y = 2x\}y=2x} be a relation on N. Find the domain …

Q2. Let X={3,4,6,8}X = \{3, 4, 6, 8\}X={3,4,6,8} . Determine whether the relation R={(x,f(x))∣x∈X,f(x)=x2+1}R = \{(x, f(x)) \mid x \in X, f(x) = x^2 + 1\}R={(x,f(x))∣x∈X,f(x)=x2+1} is a function from …

Q3. Given the function f:x→x2−5x+6f : x \rightarrow x^2 - 5x + 6f:x→x2−5x+6 , evaluate

(i)

(ii)

(iii)

(iv)

Q4. A graph representing the function f(x)f(x)f(x) is given in Fig.1.16 it is …

(i)(a)

(i)(b)

(i)(c)

(i)(d)

(ii)

(iii)

(iv)

Q5. Let f(x)=2x+5f(x) = 2x + 5f(x)=2x+5 . If x≠0x \neq 0x=0 then find f(x+2)−f(2)x\dfrac{f(x+2) - f(2)}{x}xf(x+2)−f(2)​ .

Q6. A function fff is defined by f(x)=2x−3f(x) = 2x - 3f(x)=2x−3 .

(i)

(ii)

(iii)

(iv)

Q7. An open box is to be made from a square piece of …

Q8. A function fff is defined by f(x)=3−2xf(x) = 3 - 2xf(x)=3−2x . Find xxx such that …

Q9. A plane is flying at a speed of 500 km per hour. …

Q10. The data in the adjacent table depicts the length of a person …

(i)

(ii)

(iii)

(iv)

Quick check

Q1. Let $f = \{(x,y) \mid x, y \in N$ and $y = 2x\}$ be a relation on N. Find the domain …

Q2. Let $X = \{3, 4, 6, 8\}$ . Determine whether the relation $R = \{(x, f(x)) \mid x \in X, f(x) = x^2 + 1\}$ is a function from …

Q3. Given the function $f : x \rightarrow x^2 - 5x + 6$ , evaluate

Q4. A graph representing the function $f(x)$ is given in Fig.1.16 it is …

Q5. Let $f(x) = 2x + 5$ . If $x \neq 0$ then find $\dfrac{f(x+2) - f(2)}{x}$ .

Q6. A function $f$ is defined by $f(x) = 2x - 3$ .

Q8. A function $f$ is defined by $f(x) = 3 - 2x$ . Find $x$ such that …