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Class 11 Maths Notes

Relations and Functions Class 11 Notes

Complete, exam-ready notes on relations and functions: ordered pairs and the Cartesian product, relations with their domain, codomain and range, functions and their graphs, the real functions including modulus and greatest integer, algebra of real functions, and composition — written for CBSE, JEE and NEET revision.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

What is this chapter about in one line?

A function is a relation where every input in the domain maps to exactly one output, and this chapter builds real functions, their algebra and composition.

Ordered Pairs and Cartesian Product

Ordered pair and Cartesian product

An ordered pair (a, b) fixes a first and b second, so (a, b) ≠ (b, a) unless a = b. The Cartesian product A × B is the set of all ordered pairs (a, b) with a ∈ A and b ∈ B.Domain, codomain and range of a relation R ⊆ A × B: the domain is the set of first coordinates, the codomain is B, and the range is the set of second coordinates actually appearing.

n(A×B)=n(A)⋅n(B)n(A \times B) = n(A) \cdot n(B)
Size of a Cartesian product

Relations

Relation

A relation R from A to B is a subset of A × B. Writing a R b (or (a, b) ∈ R) means a is related to b under the given condition. The domain is the set of all first coordinates and the range the set of all second coordinates of the ordered pairs in R.

  • Domain: set of first entries (a) in the relation.
  • Codomain: the full set B from which second entries may come.
  • Range: the set of second entries (b) that actually appear.
  • Total number of relations from A to B with |A| = m, |B| = n is 2^(mn).

Functions and Their Graphs

Function (mapping)

A function f : A → B assigns to every element of the domain A exactly one element of the codomain B. Its graph is the set of points (x, f(x)). Every element of the domain must have an image, so a relation that leaves an input unmapped is not a function.

One image per input

A relation is a function only if each x in the domain maps to a single y. In the graph, a vertical line must touch the curve at most once (vertical line test).

Types of Functions

  • One-one (injective): distinct inputs give distinct outputs, f(x₁) = f(x₂) implies x₁ = x₂.
  • Onto (surjective): the range equals the codomain; every element of the codomain is hit.
  • Bijective: both one-one and onto — it has an inverse.
  • Horizontal line test: a one-one function's graph meets any horizontal line at most once.

Real Functions

  • Identity: f(x) = x.
  • Constant: f(x) = c for all x.
  • Polynomial: f(x) = aₙxⁿ + ... + a₁x + a₀, defined for all real x.
  • Rational: f(x) = p(x)/q(x), undefined where q(x) = 0.
  • Modulus: |x| = x for x ≥ 0 and −x for x < 0.
  • Signum: 1 for x > 0, 0 for x = 0, −1 for x < 0.
  • Greatest integer (floor): [x] = the largest integer ≤ x.
∣x∣={x,x≥0−x,x<0|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}
Modulus function (piecewise)
[x]=largest integer ≤x[x] = \text{largest integer } \le x
Greatest integer (floor) function

Algebra of Real Functions

Sum, difference, product, quotient, composition

Real functions can be combined pointwise. Sum, difference, product and quotient are taken at each x where all the involved functions are defined; the quotient is undefined where the denominator is zero. Composition applies one function to the result of another.

(f+g)(x)=f(x)+g(x),(f−g)(x)=f(x)−g(x)(f+g)(x) = f(x) + g(x),\quad (f-g)(x) = f(x) - g(x)
Sum and difference
(fg)(x)=f(x)⋅g(x),(fg)(x)=f(x)g(x), g(x)≠0(fg)(x) = f(x) \cdot g(x),\quad \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\ g(x) \neq 0
Product and quotient
(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))
Composition of functions

Solved Examples

Example: If f(x) = x² and g(x) = x + 1, find f ∘ g(3), f + g and f/g with its domain.

Solution: f ∘ g(3) = f(4) = 16. (f + g)(x) = x² + x + 1. (f/g)(x) = x²/(x + 1) with domain all real x except x = −1.

Example: Is f(x) = x² one-one on all real numbers? Is g(x) = x³ one-one?

Solution: f(x) = x² is not one-one because f(2) = f(−2) = 4. g(x) = x³ is one-one (a strictly increasing cube), meeting the horizontal line test.

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Cartesian product size

n(A×B)=n(A)⋅n(B)n(A \times B) = n(A) \cdot n(B)

Sum of functions

(f+g)(x)=f(x)+g(x)(f+g)(x) = f(x) + g(x)

Product of functions

(fg)(x)=f(x)⋅g(x)(fg)(x) = f(x) \cdot g(x)

Quotient of functions

(fg)(x)=f(x)g(x), g(x)≠0\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)},\ g(x) \neq 0

Modulus definition

∣x∣={x,x≥0−x,x<0|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}

Greatest integer function

[x]=largest integer ≤x[x] = \text{largest integer } \le x

Composition

(f∘g)(x)=f(g(x))(f \circ g)(x) = f(g(x))

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • A function maps every domain element to exactly one output; use the vertical line test.
  • Cartesian product size: n(A × B) = n(A)·n(B).
  • One-one: distinct inputs give distinct outputs; onto: range = codomain; bijective = both.
  • Rational functions are undefined where the denominator is zero.
  • Modulus |x| is v-shaped and always non-negative.
  • Composition order matters: f ∘ g means apply g first, then f.
  • The greatest integer [x] rounds down to the nearest whole number ≤ x.

FAQ

Common questions

What is the difference between a relation and a function?

A relation is any subset of A × B. A function is a special relation where every element of the domain maps to exactly one element of the codomain; a vertical line touches a function's graph at most once.

What are the domain, codomain and range of a relation?

The domain is the set of all first coordinates, the codomain is the full target set B, and the range is the set of second coordinates that actually appear in the relation.

How do you add or multiply two functions?

Pointwise: (f + g)(x) = f(x) + g(x) and (fg)(x) = f(x)·g(x). The quotient (f/g)(x) = f(x)/g(x) is valid only where g(x) ≠ 0.

What does f ∘ g(x) mean?

It is the composition f(g(x)) — apply g to x first, then feed the result into f. The order of composition matters and cannot generally be swapped.

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