Given the function $f(x) = |x|$, which of the following describes its graph?

A straight line passing through the origin with a slope of 1.

A parabola opening upwards with vertex at the origin.

For the rational function $f(x) = \frac{x^2 - 4}{x + 2}$, what is the domain?

All real numbers except $x = -2$

All real numbers except $x = 2$

All real numbers except $x = 0$

Consider the sets $P = \{x \mid x > 0\}$ and $Q = \{y \mid y

It contains pairs where the first element is positive and the second is negative.

It contains pairs where both elements are positive.

It contains pairs where the first element is positive and the second is negative.

It contains pairs where both elements are negative.

Which of the following represents the Cartesian product $A \times B$ if $A = \{1, 2\}$ and $B = \{3, 4\}$?

$\{(1, 3), (1, 4), (2, 3), (2, 4)\}$

$\{(3, 1), (4, 1), (3, 2), (4, 2)\}$

$\{(1, 1), (2, 2), (3, 3), (4, 4)\}$

$\{(1, 4), (2, 3), (3, 2), (4, 1)\}$

If $A = \{x \mid 1 \leq x \leq 3\}$ and $B = \{y \mid 2 \leq y \leq 4\}$, which of the following is the Cartesian product $A \times B$?

$\{(1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 2), (3, 3), (3, 4)\}$

$\{(1, 1), (2, 2), (3, 3), (4, 4)\}$

StudyTunnel

by BhamuIntelli Systems LLP

StudyTunnel

Loading content...

Relations and Functions

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Polynomial Functions

Accuracy: 41%

Total Attempted27

Total Correct11

Total Incorrect16

Difficulty Level:

High

Greatest Integer Function

Accuracy: 51%

Total Attempted41

Total Correct21

Total Incorrect20

Difficulty Level:

Medium

Domain and Range of Real Functions

Accuracy: 54%

Learning Objectives

Top

Learning Objectives

Understand the concept of the Cartesian product of sets, including ordered pairs, equality of ordered pairs, and the properties of Cartesian products such as $A \times B$ , $A \times \emptyset = \emptyset$ , $A \times B \neq B \times A$ , and the cardinality $n(A \times B) = n(A)n(B)$ . Interpret $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ .
Define and represent relations as subsets of $A \times B$ using roster, set-builder, and arrow diagram forms. Identify the domain, range, codomain, image, and calculate the number of possible relations as $2^{pq}$ .
Characterize functions as a special type of relation where each element in the domain is associated with exactly one element in the codomain. Represent functions as mappings and identify their domain, codomain, and range.
Perform operations on functions including addition, subtraction, multiplication, and division, defined pointwise for functions sharing the same domain.
Analyze polynomial functions defined by expressions of the form $a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$ , where $n$ is a non-negative integer and coefficients are real numbers.
Examine rational functions as ratios of two polynomial functions, ensuring the denominator is non-zero.
Explore the modulus function, which outputs the absolute value of a number, and the signum function, which indicates the sign of a number with outputs of -1, 0, or 1.
Interpret the greatest integer function, denoted $[x]$ , which returns the greatest integer less than or equal to $x$ , resulting in a step-like graph.
Determine the domain and range of real-valued functions from formulas, tables, ordered pairs, and graphs, including restrictions from square roots and denominators.
Evaluate whether a relation is a function using ordered pairs, arrow diagrams, domain coverage, and the "one and only one image" condition.

Practice, Analyze & Improve 🚀

Dont just read—test your knowledge! Unlock the Student Workspace to take chapter tests and get instant performance insights.

Deep Analytics

Peer Comparison

Track Progress

Claim Free Account100% Free • No credit card required

Revision Notes & Summary

Top

Chapter Notes

Cartesian Product of Sets

Ordered Pairs: An ordered pair $(a, b)$ is a pair of elements grouped together in a specific order.
Equality of Ordered Pairs: Two ordered pairs $(a, b)$ and $(x, y)$ are equal if and only if $a = x$ and $b = y$ .
Cartesian Product: The Cartesian product $A \times B$ $A \times B$ of two sets $A$ $A$ and $B$ $B$ is the set of all ordered pairs $(a, b)$ $(a, b)$ where $a \in A$ $a \in A$ and $b \in B$ $b \in B$ .
- If either $A$ or $B$ is empty, $A \times B = \emptyset$ .
- Generally, $A \times B \neq B \times A$ .
- Cardinality: If $n(A) = p$ and $n(B) = q$ , then $n(A \times B) = pq$ .
Examples:
- $R \times R$ represents the set of all points in a two-dimensional space.
- $R \times R \times R$ represents the set of all points in a three-dimensional space.

Relations

Definition: A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product $A \times B$ .
Domain: The set of all first elements of the ordered pairs in a relation $R$ .
Range: The set of all second elements in a relation $R$ .
Codomain: The set $B$ in a relation $R$ from $A$ to $B$ .
Number of Relations: If $n(A) = p$ and $n(B) = q$ , then the number of relations is $2^{pq}$ .

Functions

Definition: A function $f$ from a set $A$ to a set $B$ is a relation where each element of $A$ is associated with exactly one element in $B$ .
Properties:
- Domain: The set $A$ .
- Codomain: The set $B$ .
- Range: The set of images of elements of $A$ .
Real-Valued Functions: Functions where the domain and range are subsets of real numbers.

Algebra of Functions

Addition: $(f + g)(x) = f(x) + g(x)$ .
Subtraction: $(f - g)(x) = f(x) - g(x)$ .
Multiplication: $(fg)(x) = f(x)g(x)$ .
Scalar Multiplication: $(kf)(x) = kf(x)$ , where $k$ is a real number.
Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$ , provided $g(x) \neq 0$ .

Polynomial Functions

Definition: A function of the form $f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$ , where $n$ is a non-negative integer and coefficients are real numbers.

Rational Functions

Definition: A function of the form $\frac{f(x)}{g(x)}$ , where $f(x)$ and $g(x)$ are polynomial functions and $g(x) \neq 0$ .

Modulus and Signum Functions

Modulus Function: $f(x) = |x|$ , where $f(x) = x$ if $x \geq 0$ and $f(x) = -x$ if $x < 0$ .
Signum Function: $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$ .

Greatest Integer Function

Definition: $f(x) = [x]$ , the greatest integer less than or equal to $x$ .

Domain and Range of Real Functions

Finding Domain and Range: Analyze the function's formula, considering restrictions such as square roots and denominators.

Function Testing and Mapping Diagrams

Testing Relations for Functions: Use ordered pairs and arrow diagrams to ensure each element in the domain maps to exactly one element in the codomain.

Do you belong in the Top 10%?

Our analytics show that most students struggle with specific concepts in Relations and Functions. Take a free diagnostic test to generate your personal Learning DNA profile.

Practice Test – MCQs, True/False

Top

Multiple Choice Questions

It is a parabola opening upwards.

It is a straight line passing through the origin.

It is a V-shaped graph with the vertex at the origin.

It is a horizontal line.

Correct Answer: C

Solution:

The graph of the modulus function

f(x) = |x|

is V-shaped with the vertex at the origin, consisting of two linear segments:

y = x

for

x \geq 0

and

y = -x

for

x < 0

Chapter Concept:

Algebra of Functions

-3

-2

-1

Correct Answer: A

Solution:

The greatest integer function

[x]

returns the greatest integer less than or equal to

x

. For

x = -2.3

[x] = -3

Chapter Concept:

Greatest Integer Function

Experience the StudyTunnel Method

We do not just give you mock tests. We guide you through a gamified, AI-driven learning path designed to guarantee mastery.

Step 1: Practicing the Concepts

Relations and Functions

Step 2: Master Boss Exam

Locked • Full Phase Assessment

Step 3: Redemption Arena

Locked • AI Weakness Tracker

True or False

Correct Answer: True

Solution:

The operations on functions, including addition, subtraction, multiplication, and division, are defined pointwise for functions that share the same domain. This means that these operations are performed by applying them to the corresponding values of the functions at each point in their common domain.

Chapter Concept :

Algebra of Functions

Correct Answer: True

Solution:

By definition, a polynomial function is expressed as

a_0 + a_1x + a_2x^2 + ... + a_nx^n

, with

n

being a non-negative integer and the coefficients

a_0, a_1, ..., a_n

are real numbers.

Chapter Concept :

Polynomial Functions

AI Learning Assistant

StudyTunnel Methodology

1

Master the Concepts

2

Conquer Boss Exams

3

The Redemption Arena

Relations and Functions

Study Action Plan (Sorted by Difficulty)

Polynomial Functions

High

Greatest Integer Function

Medium

Domain and Range of Real Functions

Learning Objectives

Learning Objectives

Practice, Analyze & Improve 🚀

Revision Notes & Summary

Chapter Notes

Cartesian Product of Sets

Relations

Functions

Algebra of Functions

Polynomial Functions

Rational Functions

Modulus and Signum Functions

Greatest Integer Function

Domain and Range of Real Functions

Function Testing and Mapping Diagrams

Do you belong in the Top 10%?

Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Consider the function f(x)=∣x∣f(x) = |x|f(x)=∣x∣. Which of the following statements about its graph is true?

Correct Answer: C

Q2.If f(x)=[x]f(x) = [x]f(x)=[x], what is the value of f(−2.3)f(-2.3)f(−2.3)?

Correct Answer: A

Experience the StudyTunnel Method

Step 1: Practicing the Concepts

Step 2: Master Boss Exam

Step 3: Redemption Arena

True or False

Q1.The algebra of functions includes operations such as addition, subtraction, multiplication, and division, defined pointwise for functions sharing the same domain.

Correct Answer: True

Q2.A polynomial function is defined by an expression of the form a0+a1x+a2x2+...+anxna_0 + a_1x + a_2x^2 + ... + a_nx^na0​+a1​x+a2​x2+...+an​xn, where nnn is a non-negative integer and coefficients are real numbers.

Correct Answer: True

Q3.The domain of the function f(x)=x−1f(x) = \sqrt{x-1}f(x)=x−1​ is all real numbers.

Correct Answer: False

Medium

Modulus and Signum Functions

Medium

Rational Functions

Medium

Algebra of Functions

Medium

Cartesian Product of Sets

Medium

Relations

Low

Function Testing and Mapping Diagrams

Low

Functions

Low

Q3.Given the function f(x)=∣x∣f(x) = |x|f(x)=∣x∣, which of the following describes its graph?

Correct Answer: B

Unlock the Detailed Solution

Q4.For the rational function f(x)=x2−4x+2f(x) = \frac{x^2 - 4}{x + 2}f(x)=x+2x2−4​, what is the domain?

Correct Answer: B

Unlock the Detailed Solution

Q5.If f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1​, what is the simplified form of f(x)f(x)f(x) for x≠1x \neq 1x=1?

Correct Answer: A

Unlock the Detailed Solution

Q6.Consider the sets P={x∣x>0}P = \{x \mid x > 0\}P={x∣x>0} and Q={y∣y<0}Q = \{y \mid y < 0\}Q={y∣y<0}. What can be said about the Cartesian product P×QP \times QP×Q?

Correct Answer: C

Unlock the Detailed Solution

Q7.If A={1,2}A = \{1, 2\}A={1,2} and B={3,4}B = \{3, 4\}B={3,4}, which of the following is a valid subset of the Cartesian product A×BA \times BA×B?

Correct Answer: A

Unlock the Detailed Solution

Q8.Which of the following represents the Cartesian product A×BA \times BA×B if A={1,2}A = \{1, 2\}A={1,2} and B={3,4}B = \{3, 4\}B={3,4}?

Correct Answer: A