1. Chapter at a Glance
"Is the father of", "is less than", "is the square of" — these phrases link objects in pairs. Mathematics captures such links as relations, and the most important kind, where each input has exactly one output, as functions. Functions are the central idea of calculus and of almost all applied mathematics: the distance a car travels as a function of time, the cost of electricity as a function of units used, and so on. This chapter builds the concepts from Cartesian products, then studies real functions, their graphs and their algebra.
- Ordered pairs and the Cartesian product A × B.
- Relations: domain, codomain, range; number of relations.
- Functions as special relations; domain and range of real functions.
- Standard functions and their graphs: identity, constant, polynomial, rational, modulus, signum, greatest integer.
- Algebra of functions: f ± g, fg, f/g.
2. Cartesian Product
An ordered pair (a, b) has a first and second element; (a, b) = (c, d) only if a = c and b = d.
A × B = {(a, b) : a ∈ A, b ∈ B}. If n(A) = p and n(B) = q, then n(A × B) = pq. If either set is empty, A × B = ∅.
A = {1, 2}, B = {x, y, z}. Find A × B and B × A.
Solution: A × B = {(1,x), (1,y), (1,z), (2,x), (2,y), (2,z)} — 6 pairs. B × A has the same pairs reversed; in general A × B ≠ B × A.
ℝ × ℝ = ℝ² is the coordinate plane; ℝ × ℝ × ℝ is 3-D space.
3. Relations
A relation R from A to B is any subset of A × B. If (a, b) ∈ R we write a R b. The set of first elements is the domain, B is the codomain, and the set of second elements actually used is the range (range ⊆ codomain).
Since A × B has pq elements, the number of relations from A to B is 2^(pq).
A = {1, 2, 3, 4, 5, 6}. Define R on A by R = {(x, y) : y = x + 1}. Write R, its domain and range.
Solution: R = {(1,2), (2,3), (3,4), (4,5), (5,6)}. Domain {1, 2, 3, 4, 5}; range {2, 3, 4, 5, 6}; codomain A.
4. Functions
A relation f from A to B is a function if every element of A has exactly one image in B. We write f : A → B, y = f(x). A is the domain; B the codomain; {f(x) : x ∈ A} the range.
Number of functions from A (p elements) to B (q elements) = q^p.
Vertical line test: a graph in the plane represents a function of x if every vertical line meets it at most once.
Find the domain and range of f(x) = √(9 − x²).
Solution: Need 9 − x² ≥ 0 → −3 ≤ x ≤ 3. Domain [−3, 3]. Values from 0 (at x = ±3) to 3 (at x = 0): range [0, 3].
Find the domain of f(x) = (x² + 2x + 1)/(x² − 8x + 12).
Solution: Denominator (x − 2)(x − 6) ≠ 0. Domain = ℝ − {2, 6}.
5. Standard Real Functions
| Function | Rule | Domain | Range |
|---|---|---|---|
| Identity | f(x) = x | ℝ | ℝ |
| Constant | f(x) = c | ℝ | {c} |
| Polynomial | a₀ + a₁x + … + aₙxⁿ | ℝ | depends |
| Rational | p(x)/q(x) | ℝ − {zeros of q} | depends |
| Modulus | |x| = x (x ≥ 0), −x (x < 0) | ℝ | [0, ∞) |
| Signum | 1 (x > 0), 0 (x = 0), −1 (x < 0) | ℝ | {−1, 0, 1} |
| Greatest integer | [x] = greatest integer ≤ x | ℝ | ℤ |
Examples of [x]: [2.7] = 2, [−2.7] = −3, [5] = 5.
6. Algebra of Real Functions
For f, g with domain D (common domain):
| Operation | Definition | Domain |
|---|---|---|
| Sum | (f + g)(x) = f(x) + g(x) | D |
| Difference | (f − g)(x) = f(x) − g(x) | D |
| Product | (fg)(x) = f(x)g(x) | D |
| Scalar multiple | (αf)(x) = αf(x) | D |
| Quotient | (f/g)(x) = f(x)/g(x) | D − {x : g(x) = 0} |
f(x) = x², g(x) = 2x + 1. Find (f + g)(x), (fg)(x), (f/g)(x).
Solution: x² + 2x + 1; 2x³ + x²; x²/(2x + 1), x ≠ −½.
7. Solved Problems (Board Pattern)
If (x/3 + 1, y − 2/3) = (5/3, 1/3), find x and y.
Solution: x/3 + 1 = 5/3 → x = 2. y − 2/3 = 1/3 → y = 1.
A × B has 6 elements, three of which are (1, 3), (2, 5), (3, 3). Find A, B and the remaining elements.
Solution: A ⊇ {1, 2, 3}, B ⊇ {3, 5}; 3 × 2 = 6 so A = {1, 2, 3}, B = {3, 5}. Remaining: (1, 5), (2, 3), (3, 5).
f is linear with f(1) = 1 and f(2) = 3 (as in f(x) = ax + b). Find f.
Solution: a + b = 1, 2a + b = 3 → a = 2, b = −1. f(x) = 2x − 1.
8. Master Formula Table
| Result | Formula |
|---|---|
| Size of product | n(A × B) = n(A) · n(B) |
| Number of relations A → B | 2^(pq) |
| Number of functions A → B | q^p |
| Modulus | |x| = √(x²) |
| Greatest integer | [x] ≤ x < [x] + 1 |
| Quotient domain | D − {x : g(x) = 0} |
| Square-root domain | expression under √ ≥ 0 |
9. Master Summary Table
| Concept | Key test | Exam focus |
|---|---|---|
| Cartesian product | All ordered pairs | Count; recover A and B |
| Relation | Any subset of A × B | Domain, range, roster form |
| Function | Each input → exactly one output | Is it a function? |
| Domain of real function | Denominator ≠ 0; √ needs ≥ 0 | Find domain and range |
| Standard graphs | Shape recognition | Sketch |x|, sgn x, [x] |
| Algebra of functions | Combine pointwise | f + g, fg, f/g |
10. Mind Map
- A × B
- pq pairs
- Subset of A × B
- Domain, range
- One image each
- Vertical line test
- |x|, sgn, [x]
- f ± g, fg, f/g
11. Tips and Tricks
- For domain questions, list the restrictions: denominators ≠ 0, even roots ≥ 0.
- For range, express x in terms of y and see which y-values are allowed.
- In arrow diagrams, check that every element of A has exactly one arrow leaving it.
- Confusing codomain with range.
- Thinking two inputs with the same output break the function rule (they don't).
- [−2.3] = −2 (wrong) — it is −3.
12. Important One-liners
- A × B ≠ B × A unless A = B (or one is empty).
- Every function is a relation, but not every relation is a function.
- Range ⊆ codomain.
- The range of |x| is [0, ∞); of sgn x is {−1, 0, 1}.
- Number of functions from a p-set to a q-set is q^p.
13. Exam-pattern Questions with Detailed Answers
If n(A) = 3 and n(B) = 2, the number of relations from A to B is:
- 6
- 8
- 32
- 64
Show answer
2^(3×2) = 64.
Answer: (D)
Is {(2, 1), (5, 1), (8, 1), (11, 1)} a function? Find its domain and range.
Show answer
Yes — each first element appears once. Domain {2, 5, 8, 11}; range {1}.
Answer: Yes; domain {2,5,8,11}, range {1}
Find [−4.6] + [4.6].
Show answer
−5 + 4 = −1.
Answer: −1
Find the domain and range of f(x) = 1/(2 − x).
Show answer
Domain ℝ − {2}. y = 1/(2 − x) → x = 2 − 1/y, defined for y ≠ 0. Range ℝ − {0}.
Answer: ℝ − {2}; ℝ − {0}
Find the range of f(x) = |x − 1|.
Show answer
Modulus is always ≥ 0 and takes every non-negative value: [0, ∞).
Answer: [0, ∞)
Let A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}. Find A × (B ∩ C) and (A × B) ∩ (A × C).
Show answer
B ∩ C = {4}. A × {4} = {(1,4), (2,4), (3,4)}. (A × B) ∩ (A × C) gives the same set.
Answer: {(1,4), (2,4), (3,4)} for both
Find the domain of f(x) = √(x − 1) + √(5 − x).
Show answer
x ≥ 1 and x ≤ 5 → [1, 5].
Answer: [1, 5]
If f(x) = x² − 3x + 2, find f(f(1)) and the values of x where f(x) = 0.
Show answer
f(1) = 0, f(0) = 2. Zeros: (x − 1)(x − 2) = 0 → 1, 2.
Answer: f(f(1)) = 2; x = 1, 2
Find the range of f(x) = x/(1 + x²).
Show answer
y(1 + x²) = x → yx² − x + y = 0. For real x (y ≠ 0): discriminant 1 − 4y² ≥ 0 → −½ ≤ y ≤ ½; y = 0 occurs at x = 0. Range [−½, ½].
Answer: [−1/2, 1/2]
Let R = {(a, b) : a, b ∈ ℕ, a + 3b = 12}. Write R in roster form and find its domain and range.
Show answer
b = 1 → a = 9; b = 2 → a = 6; b = 3 → a = 3; b = 4 → a = 0 (not in ℕ). R = {(9,1), (6,2), (3,3)}. Domain {3, 6, 9}; range {1, 2, 3}.
Answer: R = {(9,1), (6,2), (3,3)}
Find the domain and range of f(x) = √(x² − 4).
Show answer
x² ≥ 4 → x ≤ −2 or x ≥ 2: domain (−∞, −2] ∪ [2, ∞). Values √(x² − 4) take every value ≥ 0: range [0, ∞).
Answer: (−∞, −2] ∪ [2, ∞); [0, ∞)
f(x) = √x and g(x) = x are defined on non-negative reals. Find (f + g), (f − g), (fg) and (f/g) with domains.
Show answer
(f + g)(x) = √x + x, (f − g)(x) = √x − x, (fg)(x) = x^(3/2): all on [0, ∞). (f/g)(x) = 1/√x on (0, ∞) (g(0) = 0 excluded).
Answer: As above
14. Quick Revision Summary
- A × B: all ordered pairs; n(A × B) = pq.
- Relation: any subset of A × B; 2^(pq) relations.
- Function: each element of the domain has exactly one image; q^p functions.
- Domain: avoid zero denominators and negative even roots.
- Know graphs of identity, constant, |x|, sgn x, [x]; combine functions pointwise.