Mathematics · Class 11 Science
Ch 2Relations and Functions — Class 11 Mathematics, concept-first.
In earlier work with sets we treated the elements of a set as an unordered collection — and denote exactly the same set. Many situations, however, need us to combine two objects while keeping track of which one came first.
Key concepts
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Cartesian Product of the Reals (R×R and R×R×R)
Taking in the Cartesian-product definition gives , written , which is identified with the familiar coordinate plane — every ordered pair of reals names exactly one point, and every point names exactly one ordered pair.
Most relevant Q&A
In previous exams
How often this chapter’s concepts have been examined — real appearance data, never estimated.
Chapter contents
The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.
Ordered Pairs and Cartesian Product of Sets
In earlier work with sets we treated the elements of a set as an unordered collection — and denote exactly the same set.
Cartesian Product of the Reals — R×R and R×R×R
Taking , the set of real numbers, in the definition of the previous section gives one of the most important sets in the whole of mathematics.
Relations — Definition, Pictorial Diagrams, Domain, Co-domain and Range
Relation. Let and be two non-empty sets. A relation from to is any subset of the Cartesian product . If , we say is related to under , and write .
Functions as a Special Kind of Relation; Pictorial Representation
Function. A relation from a non-empty set to a non-empty set is called a function (or mapping) from to if every element of appears as the first component of exactly one ordered pair of — that is, for…
Real Valued Functions — Domain and Range
A function is called a real valued function if its co-domain is a subset of (usually itself), so that every output is a real number.
Standard Functions I — Constant, Identity, Polynomial and Rational Functions, with Graphs
Constant function. defined by for every , where is a fixed real number, is called a constant function. Domain ; range , a single value.
Standard Functions II — Modulus, Signum and Greatest Integer Functions, with Graphs
Modulus (absolute value) function. defined by gives the numerical (non-negative) value of , discarding its sign. Domain ; range , since is never negative but can be made as large as desired.
Standard Functions III — Exponential and Logarithmic Functions, with Graphs
Exponential function. For a fixed real number , , the function defined by is called an exponential function with base .
Algebra of Real Functions — Sum, Difference, Product and Quotient
Given two real functions and , each with its own domain, new functions can be built from them by combining their output values pointwise, at every common to both domains.
Summary
- An ordered pair keeps track of order: iff and . - The Cartesian product ; for finite sets; generally . - (written ) models the coordinate plane; (written ) models 3-D space.
Sample & Board Papers
Sample papers and previous-year board questions for this subject.
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- Q28What does the Cartesian product $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$ represent geometrically? Verify that the point $(2, -1, 3)$…Free
- Q29Find the domain and range of the real function $f(x) = \dfrac{1}{\sqrt{x - 3}}$.Preview
- Q30If $f(x) = 2x - 1$, find the value(s) of $x$ for which $f(x) = f(-x)$.Preview
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- Example 1If $(x+1,\ y-2) = (3,\ 1)$, find the values of $x$ and $y$.Free
- Example 2If $A = \{1, 2\}$ and $B = \{3, 4, 5\}$, find $A \times B$ and $n(A \times B)$.Free
- Example 3If $n(A) = 3$ and $n(B) = 4$, find the total number of relations that can be defined from $A$ to $B$.Free
- Example 4A relation $R$ is defined on the set $A = \{1, 2, 3, 4, 5\}$ by $R = \{(x, y) : x, y \in A,\ y = x + 1\}$. Write $R$ in roster form and find…Preview
- Example 5Find the domain and range of the real function $f(x) = \dfrac{1}{x - 3}$.Preview
- Example 6Find the domain and range of the real function $f(x) = \sqrt{9 - x^2}$.Preview
- Example 7Describe the graph of the signum function and state its domain and range.Preview
- Example 8If $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, find $(f+g)(x)$, $(f-g)(x)$, $(fg)(x)$ and $\left(\dfrac{f}{g}\right)(x)$, stating the domain of th…Preview
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- Q9If $A = \{a, b\}$ and $B = \{1, 2, 3\}$, find $A \times B$ and $B \times A$. Are the two sets equal?Free
- Q10If $n(A) = 5$ and $n(B) = 3$, find $n(A \times B)$ and $n(B \times A)$.Free
- Q11If $A \times B = \{(1,x), (1,y), (2,x), (2,y), (3,x), (3,y)\}$, find the sets $A$ and $B$.Preview
- Q12If $A = \{x : x \in \mathbb{N},\ x < 3\}$, find $A \times A \times A$ and the number of elements it contains.Preview
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- Q13Let $A = \{1, 2, 3, 4\}$ and let $R$ be the relation on $A$ defined by $R = \{(a,b) : a, b \in A,\ a \text{ divides } b\}$. Write $R$ in ros…Free
- Q14Determine the domain and range of the relation $R = \left\{\left(x, \dfrac{1}{x}\right) : x \text{ is a positive integer less than } 6\right…Preview
- Q15If $R = \{(x, y) : x, y \in \mathbb{N},\ x + y = 10\}$, write $R$ in roster form and find its domain and range.Preview
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- Q16Determine which of the following relations are functions from $A$ to $B$. Give reasons. (i) $\{(1,2), (1,3), (2,3), (4,5)\}$ (ii) $\{(1,4),…Free
- Q17Find the domain and range of the real function $f(x) = \sqrt{x - 4}$.Free
- Q18Find the domain of the real function $f(x) = \dfrac{x^2 + 3x + 5}{x^2 - 5x + 4}$.Preview
- Q19A function $f : \mathbb{R} \to \mathbb{R}$ is defined by $f(x) = 3x - 5$. Find $f(0)$, $f(-1)$, the value of $x$ for which $f(x) = 10$, and…Preview
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- Q20Find the domain and range of the modulus function $f(x) = |x|$.Free
- Q21Evaluate $[3.5]$, $[-2.3]$, $[7]$ and $[-7]$, where $[\,\cdot\,]$ denotes the greatest integer function.Free
- Q22If $f(x) = |x - 2| + |x + 2|$, find $f(-3)$, $f(0)$ and $f(3)$.Preview
- Q23Find the domain and range of the exponential function $f(x) = e^x$ and the logarithmic function $g(x) = \log_e x$, and state how the two fun…Preview
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- Q24If $f(x) = x + 1$ and $g(x) = 2x - 3$, find $(f+g)(2)$, $(f-g)(3)$, $(fg)(1)$ and $\left(\dfrac{f}{g}\right)(4)$.Free
- Q25If $f(x) = \sqrt{x}$ and $g(x) = \sqrt{4 - x}$, find the domain of $f + g$, $f - g$ and $fg$.Free
- Q26If $f(x) = x^2 - 1$ and $g(x) = x + 1$, find $\left(\dfrac{f}{g}\right)(x)$ in simplified form, stating its domain.Preview
- Q27If $f(x) = x^2 + 2x$ and $g(x) = x + 1$, find the domain and rule of $\left(\dfrac{f}{g}\right)(x)$. Is $x = -1$ in its domain?Preview