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MathematicsRelations and FunctionsJEE
  1. 1. Relations
  2. 2. Reflexive · Symmetric · Transitive
  3. 3. Equivalence
  4. 4. Functions
  5. 5. Composition
  6. 6. One-One & Onto
  7. 7. Bijections & Counting
  8. 8. Inverse
  9. 9. Wavy Curve
  10. 10. Domain & Range
  11. 11. Modulus
  12. 12. Trig Graphs
  13. 13. Trig Domains & Ranges
  14. 14. Inverse Trig
  15. 15. Exp & Log
  16. 16. [x], {x}, sgn
  17. 17. Even & Odd
  18. 18. Periodic
  19. 19. Functional Equations
  20. 20. Transformations
Relations and Functions · Part 4 of 20

Functions: Definition, Intervals and Domain

Intervals first, then functions: a relation where every input has exactly one output. This part covers interval notation and algebra, the definition of a function, the vertical line test, the largest domain, and explicit, implicit, bounded and piecewise functions.

Builds on: Part 1 · Relations: Definition, Domain and Range.

Functions: Definition, Intervals and DomainVideo coming soon
Intervals

Ends in or out

(a, b) leaves both ends out (hollow dots); [a, b] includes both (filled dots); [a, b) and (a, b] are half-open. All have length b − a. For A = (−3, 2] and B = (−1, 5]: A ∩ B = (−1, 2], A ∪ B = (−3, 5], A − B = (−3, −1], B − A = (2, 5].

Four number lines with open and closed ends.
Check each end separately.
Six arrow diagrams labelled (a) to (f).
Check arrows leaving A, not arriving in B.
Definition

Function or not?

A function gives every element of A exactly one image. (a) leaves 3 out and (b) gives 1 two images, so neither is a function. Shared images (d), a constant map (e) and unused elements of B are all allowed.

Vertical line test

x = y² is not a function

A curve is the graph of a function of x exactly when no vertical line meets it twice. x = 4 meets the sideways parabola at (4, 2) and (4, −2).

Sideways parabola cut twice by a vertical line.
y = x³ − 3x passes; x = y² fails.
Step-and-ramp graph of parking charges.
Hollow dot: value not taken there.
Piecewise

Parking charges

f(t) = 50 for 0 < t ≤ 3, 80 for 3 < t ≤ 5, and 80 + 30(t − 5) for t > 5. At t = 3 the filled dot belongs to the ₹50 piece.

Summary

Key formulas

Intervals as subsets of R
Intervals as subsets of R
Worked examples

From the video

1. A = (−3, 2], B = (−1, 5]. Find A ∩ B, A ∪ B, A − B, B − A.

(−1, 2], (−3, 5], (−3, −1], (2, 5].

2. Largest domain of √x.

[0, ∞).

3. Write the parking charge (₹50 up to 3 h, ₹80 for 3–5 h, then ₹30 per extra hour) as a function.

f(t) = 50 (t ≤ 3); 80 (3 < t ≤ 5); 80 + 30(t − 5) (t > 5).

JEE-style question

Your turn

A = {1, 2, 3}, B = {4, 5}. Which of these is a function from A to B?

(a){(1, 4), (2, 5)}
(b){(1, 4), (1, 5), (2, 4), (3, 5)}
(c){(1, 5), (2, 5), (3, 5)}
(d){(1, 4), (2, 5), (3, 6)}
Show the answer and the traps

{(1, 5), (2, 5), (3, 5)}: every input, one output each. Option (c), a constant function.

(a) leaves 3 without an image, and (b) gives 1 two images.

(d) sends 3 to 6, which is not in B.

Watch out

Common mistakes

Getting interval ends wrong in A − B−1 ∉ B, so it stays in A − B (closed); 2 ∈ A, so it leaves B − A (open).
Rejecting a function because two inputs share an outputOnly one input with two outputs breaks the definition.
Using the horizontal line to test for a functionThe vertical line test decides function or not.
Practice

Try these

1. Write {x : −3 ≤ x < 5} as an interval and give its length.

[−3, 5); length 8.

2. A = [−2, 4), B = (1, 6]. Find A ∩ B and A − B.

(1, 4) and [−2, 1].

3. From {1, 2, 3} to {a, b}: R₁ = {(1, a), (2, a)}, R₂ = {(1, a), (2, b), (3, a)}, R₃ = {(1, a), (1, b), (2, a), (3, b)}. Which are functions?

Only R₂.

4. Largest domain of f(x) = √(4 − x).

(−∞, 4].

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