Lesson 10 of 10 · 15 min
Chapter review
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Relations and functions with JEE-style questions
NCERT relations and functions explained step by step
Must-know facts
18 facts
- 01(a, b) = (c, d) exactly when a = c and b = d.
- 02A × B = {(a, b) : a ∈ A, b ∈ B}; A × φ = φ.
- 03n(A × B) = n(A) · n(B).
- 04A × B ≠ B × A in general, but the two have the same number of elements.
- 05A × (B ∩ C) = (A × B) ∩ (A × C) and A × (B ∪ C) = (A × B) ∪ (A × C).
- 06R × R is the plane; A × A × A is a set of ordered triplets.
- 07A relation from A to B is any subset of A × B; there are 2^(pq) of them when n(A) = p and n(B) = q.
- 08Domain: first entries. Range: second entries. Codomain: B. Range ⊂ codomain.
- 09A function gives every element of its domain exactly one image; no two pairs share a first entry.
- 10There are qᵖ functions from a set of p elements to a set of q elements.
- 11Real valued function: range in R. Real function: domain and range both in R.
- 12Identity: domain R, range R. Constant c: domain R, range {c}.
- 13x²: range [0, ∞). x³: range R. 1/x: domain and range R − {0}.
- 14|x|: domain R, range [0, ∞); V-shaped graph with vertex at the origin.
- 15Signum: domain R, range {−1, 0, 1}; sgn 0 = 0.
- 16[x] = n ⇔ n ≤ x < n + 1; range Z; [−2.4] = −3.
- 17Natural domain: denominator ≠ 0 and expression under √ ≥ 0.
- 18f/g is defined only where g(x) ≠ 0.
Common traps
Where marks are lost
Formulas
10 to know
Cartesian product
A × B = {(a, b) : a ∈ A, b ∈ B}
A × φ = φ; ordered pairs, so order matters.
Size of a product
n(A × B) = n(A) · n(B)
If n(A) = p and n(B) = q, A × B has pq pairs.
Number of relations
2^(pq)
Relations from A to B are the subsets of A × B.
Number of functions
qᵖ
Each of the p elements of A picks one of the q elements of B.
Identity and constant
f(x) = x; f(x) = c
Ranges R and {c}.
Modulus function
|x| = x if x ≥ 0; |x| = −x if x < 0
Range [0, ∞).
Signum function
sgn x = 1 if x > 0; 0 if x = 0; −1 if x < 0
Range {−1, 0, 1}; equals |x|/x for x ≠ 0.
Greatest integer function
[x] = n ⇔ n ≤ x < n + 1, n ∈ Z
Range Z; [x] ≤ x < [x] + 1.
Sum, difference, scalar multiple
(f ± g)(x) = f(x) ± g(x); (αf)(x) = α f(x)
On the common domain.
Product and quotient
(fg)(x) = f(x) g(x); (f/g)(x) = f(x)/g(x)
The quotient needs g(x) ≠ 0.
Key terms
16 terms
- Ordered pair
- Two objects in a fixed order, (a, b), where swapping them gives a different pair.
- Cartesian product
- A × B, the set of all ordered pairs with first entry from A and second from B.
- Ordered triplet
- Three objects in a fixed order, (a, b, c); A × A × A is a set of them.
- Relation
- Any subset of A × B, picked out by a rule linking first and second entries.
- Image
- For a pair (x, y) in a relation or function, y is the image of x.
- Preimage
- For f(x) = y, x is a preimage of y.
- Domain
- The set of first entries; for a function f : A → B, the whole of A.
- Codomain
- The set B that images are allowed to come from.
- Range
- The set of images actually produced; always a subset of the codomain.
- Function
- A relation giving every element of its domain exactly one image.
- Real function
- A function whose domain and range are both subsets of R.
- Polynomial function
- a₀ + a₁x + … + aₙxⁿ with real coefficients and n a non-negative integer.
- Rational function
- A quotient f(x)/g(x) of polynomial functions, defined where g(x) ≠ 0.
- Modulus function
- |x|, the distance of x from 0: x for x ≥ 0 and −x for x < 0.
- Signum function
- Gives 1, 0 or −1 according as x is positive, zero or negative.
- Greatest integer function
- [x], the largest integer that does not exceed x.