Points to Remember: Relation and Function | Class 10th Mathematics

Points to Remember: Relation and Function

Key Concepts

• The Cartesian Product of \(A\) with \(B\) is defined as \(A \times B = \{(a, b) \mid a \in A, b \in B\}\).
• A relation R from \(A\) to \(B\) is always a subset of \(A \times B\). That is, \(R \subseteq A \times B\).
• A relation R from \(X\) to \(Y\) is a function if for every \(x \in X\) there exists only one \(y \in Y\).
• A function can be represented by:
  • (i) an arrow diagram
  • (ii) a tabular form
  • (iii) a set of ordered pairs
  • (iv) a graphical form
• Some types of functions:
  • (i) One-one function
  • (ii) Onto function
  • (iii) Many-one function
  • (iv) Into function
• Identity function: \(f(x) = x\)
• Reciprocal function: \(f(x) = \frac{1}{x}\)
• 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: \(f(x) = ax^3 + bx^2 + cx + d, a \neq 0\)
• For three non-empty sets \(A\), \(B\) and \(C\), if \(f : A \to B\) and \(g : B \to C\) are two functions, then the composition of \(f\) and \(g\) is a function \(g \circ f : A \to C\) will be defined as \((g \circ f)(x) = g(f(x))\) for all \(x \in A\).
• If \(f\) and \(g\) are any two functions, then in general, \(f \circ g \neq g \circ f\).
• If \(f\), \(g\) and \(h\) are any three functions, then \(f \circ (g \circ h) = (f \circ g) \circ h\).

10th Maths Unit 1: Relation and Function Exercise Problems with Solutions

Exercise: Relation and Function - Problems & Solutions

UNIT EXERCISE

Question 1

If the ordered pairs \( (x^2 - 3x, y^2 + 4y) \) and (-2, 5) are equal, then find \(x\) and \(y\).

Question 2

The cartesian product \(A \times A\) has 9 elements among which (–1, 0) and (0,1) are found. Find the set A and the remaining elements of \(A \times A\).

Question 3

Given that \(f(x)\) =

(i) \(f(0)\)

(ii) \(f(3)\)

(iii) \(f(a + 1)\) in terms of \(a\). (Given that \(a \ge 0\))

Question 4

Let A = {9,10,11,12,13,14,15,16,17} and let \(f: A \to N\) be defined by \(f(n)\) = the highest prime factor of \(n \in A\). Write f as a set of ordered pairs and find the range of f.

Question 5

Find the domain of the function \( f(x) = \sqrt{1+\sqrt{1-\sqrt{1-x^2}}} \)

Question 6

If \(f(x) = x^2\), \(g(x) = 3x\) and \(h(x) = x - 2\), Prove that \((f \circ g) \circ h = f \circ (g \circ h)\).

Question 7

Let A = {1, 2} and B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify whether \(A \times C\) is a subset of \(B \times D\).

Question 8

If \(f(x) = \frac{x-1}{x+1}, x \neq -1\) show that \(f(f(x)) = -\frac{1}{x}\) provided \(x \neq 0\).

Question 9

The functions f and g are defined by \(f(x) = 6x + 8\); \(g(x) = \frac{x-2}{3}\).

(i). Calculate the value of \(gg(\frac{1}{2})\)

(ii) Write an expression for \(gf(x)\) in its simplest form.

Question 10

Write the domain of the following real functions:

(iv) \(h(x) = x + 6\)

Answers

1. 1,2 and -5, 1
2. {-1, 0, 1} , {(-1, -1),(-1, 1),(0, -1),(0, 0),(1, -1),(1, 0),(1, 1)}
3. (i) 4 (ii) \(\sqrt{2}\) (iii) \(\sqrt{a}\)
4. {(9, 3),(10, 5),(11, 11),(12, 3),(13, 13),(14, 7),(15, 5),(16, 2),(17, 17)} , {2,3,5,11,13,17}
5. {–1, 0, 1}
6. (i) -5/6 (ii) 2(x + 1)
7. (i) R - {9} (ii) R (iii) [2, ∞) (iv) R

10th Maths Unit 1: Relation and Function MCQs with Solutions

Multiple choice questions - Relation and Function | Mathematics

Mathematics: Relation and Function: Multiple choice questions with answers / choose the correct answer with answers - Maths Book back 1 mark questions and answers with solution for Exercise Problems.

Question 1

If \(n(A \times B) = 6\) and \(A = \{1, 3\}\) then \(n(B)\) is

• (1) 1
• (2) 2
• (3) 3
• (4) 6

Solution

Question 2

A = {a,b, p}, B = {2, 3}, C = {p,q,r,s} then \(n[(A \cup C) \times B]\) is

• (1) 8
• (2) 20
• (3) 12
• (4) 16

Solution

Question 3

If A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8} then state which of the following statement is true.

• (1) \((A \times C) \subset (B \times D)\)
• (2) \((B \times D) \subset (A \times C)\)
• (3) \((A \times B) \subset (A \times D)\)
• (4) \((D \times A) \subset (B \times A)\)

Solution

Question 4

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) 3
• (2) 2
• (3) 4
• (4) 8

Solution

Question 5

The range of the relation \(R = \{(x, x^2) | x \text{ is a prime number less than 13}\}\) is

• (1) {2,3,5,7}
• (2) {2,3,5,7,11}
• (3) {4,9,25,49,121}
• (4) {1,4,9,25,49,121}

Solution

Question 6

If the ordered pairs (a + 2, 4) and (5, 2a + b) are equal then (a,b) is

• (1) (2, –2)
• (2) (5,1)
• (3) (2,3)
• (4) (3, –2)

Solution

Question 7

Let \(n(A) = m\) and \(n(B) = n\) then the total number of non-empty relations that can be defined from A to B is

• (1) \(m^n\)
• (2) \(n^m\)
• (3) \(2^{mn} – 1\)
• (4) \(2^{mn}\)

Solution

Question 8

If {(a, 8),(6,b)} represents an identity function, then the value of a and b are respectively

• (1) (8,6)
• (2) (8,8)
• (3) (6,8)
• (4) (6,6)

Question 9

Let A = {1, 2, 3, 4} and B = {4, 8, 9,10}. A function \(f: A \rightarrow B\) given by f = {(1, 4),(2, 8),(3, 9),(4,10)} is a

• (1) Many-one function
• (2) Identity function
• (3) One-to-one function
• (4) Into function

Question 10

If \(f(x) = 2x^2\) and \(g(x) = \frac{1}{3x}\) then \(f \circ g\) is

Ans: (3)

Solution

Question 11

If \(f: A \rightarrow B\) is a bijective function and if \(n(B) = 7\), then \(n(A)\) is equal to

• (1) 7
• (2) 49
• (3) 1
• (4) 4

Question 12

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) {0,2,3,4,5}
• (2) {–4,1,0,2,7}
• (3) {1,2,3,4,5}
• (4) {0,1,2}

Solution

Question 13

Let \(f(x) = \) then

• (1) \(f(xy) = f(x).f(y)\)
• (2) \(f(xy) \geq f(x).f(y)\)
• (3) \(f(xy) \leq f(x).f(y)\)
• (4) None of these

Solution

Question 14

If g = {(1,1),(2, 3),(3, 5),(4, 7)} is a function given by \(g(x) = \alpha x + \beta\) then the values of a and b are

• (1) (–1,2)
• (2) (2, –1)
• (3) (–1, –2)
• (4) (1,2)

Solution

Question 15

\(f(x) = (x + 1)^3 - (x - 1)^3\) represents a function which is

• (1) linear
• (2) cubic
• (3) reciprocal
• (4) quadratic

Solution

Answers Summary

Tags: Relation and Function | Mathematics, 10th Mathematics: UNIT 1: Relation and Function

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Identifying Graphs of Linear, Quadratic, Cubic, and Reciprocal Functions

10th Mathematics : UNIT 1 : Relation and Function

Identifying the graphs of Linear, Quadratic, Cubic and Reciprocal functions

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Graphs provide visualization of curves and functions. Hence, graphs help a lot in understanding the concepts in a much efficient way.

In this section, we will be discussing about the identification of some of the functions through their graphs. In particular, we discuss graphs of Linear, Quadratic, Cubic and Reciprocal functions.

1. Linear Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = mx + c$, $m \neq 0$ is called a linear function. Geometrically this represents a straight line in the graph.

Some Specific Linear Functions and their graphs are given below.

2. Modulus or Absolute valued Function

A function $f: \mathbb{R} \rightarrow [0, \infty)$ defined by $f(x) = |x|$.

Note

• Modulus function is not a linear function but it is composed of two linear functions x and –x.
• Linear functions are always one-one functions and has applications in Cryptography as well as in several branches of Science and Technology.

3. Quadratic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^2 + bx + c$, $(a \neq 0)$ is called a quadratic function.

Some specific quadratic functions and their graphs:

The equations of motion of a particle travelling under the influence of gravity is a quadratic function of time. These functions are not one – one. (Why?)

4. Cubic Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = ax^3 + bx^2 + cx + d$, $(a \neq 0)$ is called a cubic function. The graph of $f(x) = x^3$ is shown in Fig.1.48.

5. Reciprocal Function

A function $f: \mathbb{R} - \{0\} \rightarrow \mathbb{R}$ defined by $f(x) = \frac{1}{x}$ is called a reciprocal function (Fig.1.49).

6. Constant Function

A function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = c$, for all $x \in \mathbb{R}$ is called a constant function (Fig.1.50).

Composition of Functions: Definition, Illustration, Examples, and Solutions

Composition of Functions - Definition, Illustration, Example, Solution

Introduction to Composition of Functions

When a car driver depresses the accelerator pedal, it controls the flow of fuel which in turn influences the speed of the car. Likewise, the composition of two functions is a kind of ‘chain reaction’, where the functions act upon one after another.

Fig. 1.40

We can explain this further with the concept that a function is a ‘process’. If f and g are two functions then the composition g(f (x)) is formed in two steps.

1. Feed an input (say x) to f;
2. Feed the output f(x) to g to get g(f (x)) and call it gf(x).

Fig. 1.41

Illustration

Consider the set A of all students, who appeared in class X of Board Examination. Each student appearing in the Board Examination is assigned a roll number. In order to have confidentiality, the Board arranges to deface the roll number of each student and assigns a code number to each roll number.

Let A be the set of all students appearing for the board exam. B ⊆ N be the set all roll numbers and C ⊆ N be the set of all code numbers. This gives rise to two functions f: A → B and g: B → C given by b = f (a) be the roll number assigned to student a, c = g(b) be the code number assigned to roll number b, where a ∈ A , b ∈ B and c ∈ C.

We can write c = g(b) = g(f (a)).

Thus, by the combination of these two functions, each student is eventually attached a code number. This idea leads to the following definition.

Definition

Let f : A → B and g : B → C be two functions (Fig.1.42). Then the composition of f and g denoted by g o f is defined as the function g o f (x ) = g( f (x )) for all x ∈ A.

Fig. 1.42

Example 1.20

Find f o g and g o f when f (x) = 2x + 1 and g(x) = x² – 2

Solution

f (x) = 2x + 1 , g(x) = x² – 2

f o g(x) = f (g(x)) = f (x² − 2) = 2(x² − 2) + 1 = 2x² – 3

g o f (x) = g(f (x)) = g(2x + 1) = (2x + 1)² − 2 = 4x² + 4x – 1

Thus f o g = 2x² − 3, g o f = 4x² + 4x − 1. From the above, we see that f o g ≠ g o f .

Note

Generally, f o g ≠ g o f for any two functions f and g. So, composition of functions is not commutative.

Example 1.21

Represent the function f(x) = √(2x² − 5x + 3) as a composition of two functions.

Solution

We set f₁(x) = 2x² − 5x + 3 and f₂(x) = √x

Then,

Example 1.22

If f (x) = 3x − 2 , g(x) = 2x + k and if f o g = g o f , then find the value of k.

Solution

f(x) = 3x − 2 , g(x) = 2x + k

f o g(x) = f (g(x)) = f (2x + k) = 3(2x + k) − 2 = 6x + 3k – 2

Thus, f o g(x) = 6x + 3k – 2.

g o f (x) = g(3x − 2) = 2(3x − 2) + k

Thus, g o f (x) = 6x − 4 + k.

Given that f o g = g o f

Therefore, 6x + 3k − 2 = 6x − 4 + k

6x − 6x + 3k − k = −4 + 2 ⇒ 2k = −2 ⇒ k = −1

Example 1.23

Find k if f o f (k) = 5 where f (k) = 2k – 1.

Solution

f o f (k) = f (f (k))

= 2(2k − 1) − 1 = 4k − 3

Thus, f o f (k) = 4k – 3

But, it is given that f o f (k) = 5

Therefore 4k - 3 = 5 ⇒ 4k = 8 ⇒ k = 2.

Composition of three functions

Let A, B, C, D be four sets and let f : A → B , g : B → C and h : C → D be three functions (Fig.1.43). Using composite functions f o g and g o h, we get two new functions like (f o g) o h and f o (g o h).

Fig. 1.43

We observed that the composition of functions is not commutative. The natural question is about the associativity of the operation.

Note

Composition of three functions is always associative. That is, f o (g o h) = (f o g) o h

Example 1.24

If f(x) = 2x + 3, g(x) = 1 − 2x and h(x) = 3x. Prove that f o (g o h) = (f o g) o h

Solution

f(x) = 2x + 3 , g(x) = 1 − 2x , h(x) = 3x

Now, (f o g)(x) = f (g(x)) = f (1 − 2x) = 2(1 − 2x) + 3 = 5 − 4x

Then, (f o g) o h(x) = (f o g)(h(x)) = (f o g)(3x) = 5 − 4(3x) = 5 − 12x ……… (1)

(g o h)(x) = g(h(x)) = g(3x) = 1 − 2(3x) = 1 − 6x

So, f o (g o h)(x) = f (1 − 6x) = 2(1 − 6x ) + 3 = 5 − 12x ……… (2)

From (1) and (2), we get (f o g) o h = f o (g o h)

Example 1.25

Find x if gff(x) = fgg(x), given f (x) = 3x + 1 and g(x) = x + 3.

Solution

gff(x) = g [f {f (x)}] (This means “g of f of f of x”)

= g [ f (3x +1)] = g [ 3(3x +1)+1] = g (9x + 4)

g (9x + 4) = [ (9x + 4) + 3] = 9x + 7

fgg(x) = f [g {g (x)}] (This means “f of g of g of x”)

= f [ g (x + 3)] = f [ (x + 3) + 3] = f (x + 6)

f (x + 6) = [ 3(x + 6) + 1 ] = 3x + 19

These two quantities being equal, we get 9x + 7 = 3x + 19. Solving this equation we obtain x = 2.

10th Maths Unit 1 Exercise 1.4: Types of Functions Solutions

Exercise 1.4: Types of Functions - Problem Questions with Answer, Solution | Mathematics

Question 1

Determine whether the graph given below represent functions. Give reason for your answers concerning each graph.

Solution:

Question 2

Let \(f : A \rightarrow B\) be a function defined by \(f(x) = \frac{x}{2} - 1\), where \(A = \{2, 4, 6, 10, 12\}\), \(B = \{0, 1, 2, 4, 5, 9\}\). Represent \(f\) by

• (i) set of ordered pairs;
• (ii) a table;
• (iii) an arrow diagram;
• (iv) a graph

Solution:

Question 3

Represent the function \(f = \{(1, 2), (2, 2), (3, 2), (4, 3), (5, 4)\}\) through

• (i) an arrow diagram
• (ii) a table form
• (iii) a graph

Solution:

Question 4

Show that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(x) = 2x - 1\) is one-one but not onto.

Solution:

Question 5

Show that the function \(f: \mathbb{N} \rightarrow \mathbb{N}\) defined by \(f(m) = m^2 + m + 3\) is one-one function.

Solution:

Question 6

Let \(A = \{1, 2, 3, 4\}\) and \(B = \mathbb{N}\). Let \(f: A \rightarrow B\) be defined by \(f(x) = x^3\) then,

• (i) find the range of \(f\)
• (ii) identify the type of function

Solution:

Question 7

In each of the following cases state whether the function is bijective or not. Justify your answer.

• (i) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 2x + 1\)
• (ii) \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(f(x) = 3 - 4x^2\)

Solution:

Question 8

Let \(A = \{-1, 1\}\) and \(B = \{0, 2\}\). If the function \(f: A \rightarrow B\) defined by \(f(x) = ax + b\) is an onto function? Find \(a\) and \(b\).

Solution:

Question 9

If the function \(f\) is defined by $$ f(x) = \begin{cases} x+2 & \text{if } x > 1 \\ 2 & \text{if } -1 \le x \le 1 \\ x-1 & \text{if } -3 < x < -1 \end{cases} $$ find the values of

• (i) \(f(3)\)
• (ii) \(f(0)\)
• (iii) \(f(-1.5)\)
• (iv) \(f(2) + f(-2)\)

Solution:

Question 10

A function \(f: [-5, 9] \rightarrow \mathbb{R}\) is defined as follows: $$ f(x) = \begin{cases} 6x+1 & \text{if } -5 \le x < 2 \\ 5x^2-1 & \text{if } 2 \le x < 6 \\ 3x-4 & \text{if } 6 \le x \le 9 \end{cases} $$ Find

• (i) \(f(-3) + f(2)\)
• (ii) \(f(7) - f(1)\)
• (iii) \(2f(4) + f(8)\)
• (iv) \(\frac{2f(-2) - f(6)}{f(4) + f(-2)}\)

Solution:

Question 11

The distance \(S\) an object travels under the influence of gravity in time \(t\) seconds is given by \(S(t) = \frac{1}{2}gt^2 + at + b\) where, (\(g\) is the acceleration due to gravity), \(a\), \(b\) are constants. Check if the function \(S(t)\) is one-one.

Solution:

Question 12

The function ‘\(t\)’ which maps temperature in Celsius (\(C\)) into temperature in Fahrenheit (\(F\)) is defined by \(t(C) = F\) where \(F = \frac{9}{5}C + 32\). Find,

• (i) \(t(0)\)
• (ii) \(t(28)\)
• (iii) \(t(-10)\)
• (iv) the value of \(C\) when \(t(C) = 212\)
• (v) the temperature when the Celsius value is equal to the Farenheit value.

Solution:

Answers

1. (i) Not a function (ii) function (iii) Not a function (iv) function

2. (i) $\{(2,0),(4,1),(6,2),(10,4),(12,5)\}$

6. (i) $\{1, 8, 27, 64\}$ (ii) one-one and into function

7. (i) Bijective function (ii) Not bijective function

8. 1,1

9. (i) 5 (ii) 2 (iii) -2.5 (iv) 1

10. (i) 2 (ii) 10 (iii) 178 (iv) -9/17

11. Yes

12. (i) 32°F (ii) 82.4°F (iii) 14°F (iv) 100°C (v) -40°